Voltage and Frequency Regulation in Interconnected Power Systems via a (1+PDD²)-(1+TI) Cascade Controller Optimized by Mirage Search Optimizer

Abstract

The combined application of Load Frequency Control (LFC) and Automatic Voltage Regulation (AVR), known as Automatic Generation Control (AGC), manages active and reactive power to ensure system stability. This study presents a novel hybrid controller with a (1+PDD²)-(1+TI) structure, optimized using the Mirage Search Optimization (MSO) algorithm. Designed for dual-area power systems, the controller enhances both LFC and AVR by coordinating voltage and frequency loops. MSO was chosen after outperforming five algorithms (ChOA, DOA, PSO, GTO, and GBO), achieving the lowest fitness value (ITSE = 0.028). The controller was tested under various challenging conditions: sudden load disturbances, stochastic variations, nonlinearities like Generation Rate Constraints (GRC) and Governor Dead Band (GDB), time-varying reference voltages, and ±20% to ±40% parameter deviations. Across all scenarios, the (1+PDD²)-(1+TI) controller consistently outperformed MSO-tuned TID, FOPID, FOPI-PIDD², (1+PD)-PID, and conventional PID controllers. It demonstrated superior performance in regulating frequency, tie-line power, and voltage, achieving approximately a 50% improvement in dynamic response. MATLAB/SIMULINK results confirm its effectiveness in enhancing overall system stability.

1. Introduction

1.1. Background

The stability of an integrated power system represents a complex, multi-variable, and multi-goal control issue, with its dynamic behavior heavily influenced by imbalances between power generation and demand. This imbalance serves as a key indicator of overall system stability. Power system stability is typically divided into three main categories: frequency stability, voltage stability, and rotor angle stability. Stability of frequency is primarily impacted by fluctuations in active power, while voltage stability is influenced by imbalances in reactive power. Maintaining frequency stability involves the system’s ability to restore the balance between generated power and load demand with minimal loss [1]. On the other hand, voltage stability refers to the system’s capability to maintain voltage levels close to their nominal values at all buses, even when subjected to disturbances [2]. To stabilize both active and reactive power, adjustments can be made through the speed governor in the Load Frequency Control (LFC) loop and the exciter in the Automatic Voltage Regulation (AVR) loop. When LFC and AVR systems are integrated, the process is commonly known as Automatic Generation Control (AGC) [3,4].

1.2. Literature Review

Initial research primarily focused on load frequency control (LFC) studies involving electric vehicles (EVs) as a means to support conventional thermal power plants in mitigating system frequency variations [5]. With the increasing integration of wind and solar photovoltaic energy, power supply and demand have become more intermittent. To address this challenge, new LFC strategies, such as the state of charge (SOC) method for EVs, have been introduced [6]. Additionally, to counter significant variations in distribution frequency and voltage, a supplementary LFC strategy involving a large number of EVs was proposed [7]. The application of ultra-batteries [8] and redox flow batteries has also been explored in both deregulated [9] and conventional [10] LFC systems. Studies analyzing EVs in combination with other renewable energy sources further highlight their effectiveness [11,12]. However, a comprehensive investigation into a unified system for Automatic Load Frequency and Voltage Control for EVs remains lacking, indicating the need for further research in this area. Fractional-order (FO) controller structures have gained significant traction in recent times due to their increased flexibility and broader range of tuning parameters. Among these, the Tilted Integral Derivative (TID) controller, which falls under the umbrella of fractional-order calculus (FOC), has been applied to address challenges in load frequency control. TID controllers are favored for their ease of tuning, higher disturbance rejection capability, and minimal impact on the plant. Consequently, many studies have proposed the TID controller as a viable alternative for tackling LFC-related issues [13,14,15]. Furthermore, references [16,17] introduce a composite controller that combines the TID and Fractional-Order Proportional integral Derivative (FOPID) controllers to leverage the strengths of both approaches. This hybrid design offers simplified tuning, improved disturbance rejection, and reduced plant sensitivity. Recently, cascaded controller structures have gained popularity over traditional control methods due to their enhanced effectiveness and superior performance. Various types of cascaded controllers have been utilized to improve frequency stability in power systems [18,19,20]. Additionally, recent LFC and AVR research has explored the strategy of combining two different controllers to further boost control performance [21,22].

To achieve an optimal dynamic response, advanced control strategies have been employed in the Egyptian power grid, including adaptive controllers [23]. Multi-area power systems experiencing communication delays have been controlled using Linear–Quadratic–Gaussian (LQG) regulators [24,25]. Fuzzy-based controllers have also been developed to autonomously regulate the voltage of synchronous generators [26,27,28]. Furthermore, artificial neural network (ANN)-based Automatic Voltage Regulators (AVRs) have been proposed in order to strengthen the ability of power systems to maintain stability during transient disturbances [2,29], and fractional-order controllers have also been implemented to enhance the effectiveness of AVR systems [30,31,32]. Despite demonstrating high efficiency in simulations, these advanced controllers often come with significant computational and structural complexity. Classical approaches, on the other hand, aim to determine suitable parameters based on linear and estimated models of the system. These methods include evolutionary and metaheuristic algorithms, for instance, Genetic Algorithms (GA) [33,34], Sine Cosine Algorithm [35,36], Particle Swarm Optimization (PSO) [37,38], Whale Optimization Algorithm [39], Imperialist Competitive Algorithm (ICA) [40,41], Teaching–Learning-Based Optimization [42], and Seeker Optimization (SO) [43,44]. In these approaches, the controller design is formulated as an optimization problem, where tuning the controller parameters leads to performance enhancement. Although these methods perform well in simulations, significantly, the AVR and Load Frequency Control (LFC) systems are inherently nonlinear and subject to parameter variations. Consequently, they are affected by both parametric and structural uncertainties, which can degrade overall system performance.

In this study, the Mirage Search Optimization (MSO) algorithm is employed to tune the controller parameters. Its performance is compared with five other optimization algorithms: Chimp Optimization Algorithm (ChOA) [45], Dingo Optimization Algorithm (DOA) [46], Particle Swarm Optimization (PSO) [47,48], Gorilla Troops Optimization (GTO) [49,50], and Whale Optimization Algorithm (GBO) [51]. The results indicate that MSO demonstrates strong competitiveness in unimodal and hybrid functions, although it performs less effectively in multimodal functions. Moreover, MSO shows promising performance in high-dimensional search spaces, making it suitable for tackling relatively complex high-dimensional optimization problems [52]. In [53], the authors introduced a FOPI-PIDD² controller structure aimed at minimizing frequency and voltage fluctuations in a two-area interconnected power system. Their results demonstrated that the FOPI-PIDD² controller outperforms both the conventional PID and TID controllers. Similarly, for load frequency control, the (1+PD)-PID controller presented in [54] showed better frequency regulation performance compared to the traditional PID controller. However, the newly proposed (1+PDD²)-(1+TI) controller surpasses both the FOPI-PIDD² and (1+PD)-PID controllers in performance. Earlier studies have highlighted that choosing suitable controller parameters is just as important as selecting the appropriate type of controller. Within this framework, evolutionary optimization methods have played a significant role in enhancing controller parameter tuning, thereby improving frequency and voltage stability.

This motivated the authors to develop a novel combined (1+PDD²)-(1+TI) regulator for the integrated LFC-AVR model in a dual-area power system, aiming to enhance both voltage and frequency stability. Until now, most research has focused primarily on implementing conventional and fractional-order (FO) controllers within the combined LFC-AVR framework. However, due to the increased complexity of the integrated LFC-AVR system, traditional controllers are no longer sufficient, especially under severe disturbances. Therefore, a (1+PDD²)-(1+TI) controller, fine-tuned using the MSO algorithm, was adopted as a secondary control strategy in both the LFC and AVR loops.

1.3. Contributions

This study introduces an innovative (1+PDD²)-(1+TI) controller configuration designed to improve system frequency and voltage stability in the presence of power disturbances. Furthermore, The MSO technique was employed to optimize the controller parameters to maintain stability under abnormal operating conditions. Unlike previous research on similar topics, this paper’s key contributions can be summarized as follows:

• The development and implementation of a robust (1+PDD²)-(1+TI) controller to improve frequency and voltage stability in a dual-area interconnected power system;
• The application of the Mirage Search Optimization (MSO) algorithm to fine-tune the parameters of the proposed controller, offering a creative and high-performance approach to LFC and AVR optimization;
• A comprehensive comparison demonstrating the exceptional effectiveness of the MSO algorithm over several advanced optimization techniques, including ChOA, DOA, PSO, GTO, and GBO;
• Validation of the proposed controller’s performance against various established control strategies from existing literature, such as TID [55,56], FOPID [57], FOPI-PIDD² [53], (1+PD)-PID, and the conventional PID [54];
• In-depth analysis of the proposed controller’s performance and robustness under multiple scenarios, including sudden load changes at t = 0, random load fluctuations, commonly exhibit nonlinearities in power systems (such as Generation Rate Constraints (GRC) and Governor Dead Band (GDB)), time-varying reference voltages in both areas, and ±20% to ±40% variations in system parameters.

The remainder of the paper is organized as follows:

• Section 2: outlines the architecture of the studied system and describes its components;
• Section 3: presents the proposed MSO algorithm with details the configuration of the designed controller;
• Section 4: discusses the simulation results under various test scenarios;
• Section 5: concludes the paper with final observations and recommendations.

2. Representation of a Two-Area Interconnected Power Network

2.1. Dual-Area Interconnected Power System

The proposed controller’s performance is assessed on a conventional two-area power system. As illustrated in Figure 1, this system features two zones with equal power generation capacity. The diagram provides a simplified depiction of the integration between the Automatic Voltage Regulator (AVR) and Load Frequency Control (LFC) loops in each area. The AVR loop is responsible for ensuring consistent output voltages of the generating units, while the LFC loop focuses on regulating frequency deviations caused by active load disturbances [58,59]. Notably, active load variations in one area affect not only the local frequency but also introduce disturbances to the interconnected area. Figure 1 also illustrates the tie-line connection between the two areas, which facilitates power exchange. A graphical model can represent the interconnection of the dual-area system, where the synchronization coefficient K₁₂ represents the graph weights [60]. This synchronization-based model reflects a consensus mechanism, with frequency deviation values serving as the states for each area. The speed of synchronization across the system is governed by the magnitude of the synchronization coefficient. The next sections will outline the details of the dynamic components of the integrated AVR-LFC scheme.

2.2. Automatic Voltage Regulation

The Automatic Voltage Regulator (AVR) system aims to decrease reactive power losses that result from discrepancies between the desired voltage and the exciter terminal voltage $E_e$. Fluctuations in a generator’s reactive power load cause fluctuations in its terminal voltage $E$. As illustrated in Figure 2, this terminal voltage is sensed through a single-phase potential transformer, yielding a measured voltage $∆V$ or $E_S$, and then measured against a predetermined reference voltage $V_{Ref}$. The generated error signal undergoes amplification to generate the required voltage $E_a$, which is used to regulate the exciter’s field, thereby adjusting the exciter’s terminal voltage. Ultimately, this influences the field current of the generator and changes the resulting induced electromotive force (emf) [60]. The schematic diagram of AVR is presented in Figure 2. A comprehensive mathematical model of the AVR unit is presented in

Table 1. AVR system parameters and configuration for Area 1 and Area 2.

Model	Transfer Function	Parameters	Nominal Values	Parameter Description
Amplifier	${\displaystyle \frac{K_A}{1+T_{A}S}}$	$K_A$, $T_A$	10, 0.1	Gains and time constants of amplifier, exciter, generator and sensor
Exciter	${\displaystyle \frac{K_E}{1+T_{E}S}}$	$K_E$, $T_E$	1, 0.4
Generator (Field CKT)	${\displaystyle \frac{K_F}{1+T_{F}S}}$	$K_F$, $T_F$	1, 1
Sensor	${\displaystyle \frac{K_S}{1+T_{S}S}}$	$K_S$, $T_S$	1, 0.01

.

2.3. Load Frequency Control

As shown in Figure 2, a power system utilizing Load Frequency Control (LFC) includes components such as a rotating mass, governor, turbine, and load demands. The core objective of the LFC mechanism is to manage dynamic load fluctuations among multiple generators, maintain frequency stability, and regulate power transfer through tie-lines [53]. Frequency deviations ($\mathsf{\Delta }f$) in each area correspond to changes in the generator’s rotor angle ($\mathsf{\Delta }\delta$). These deviations in frequency are measured, and the resulting error signal undergoes amplification and processing to yield a real-power correction signal ($\mathsf{\Delta }{P}_G$) [1]. This correction prompts the prime mover to adjust its torque output accordingly.

Three fundamental differential equations characterize the dynamic response of the LFC system by modeling the interactions between the governor, turbine, and generator or load components [1,53].

Table 2. Two-area LFC parameter values.

Model	Transfer Function	Parameters	Nominal Values Area 1	Nominal Values Area 2	Parameter Description
Governor	${\displaystyle \frac{K_g}{1+T_{g}S}}$	$K_g$, $T_g$	1, 0.08	1, 0.072	Governor, Turbine, and Load gains and time constants
Turbine	${\displaystyle \frac{K_t}{1+T_{t}S}}$	$K_t$, $T_t$	1, 0.3	1, 0.33
Load	${\displaystyle \frac{K_p}{1+T_{p}S}}$	$K_p$, $T_p$	120, 20	112.5, 25
		$K_B$	0.41	0.37	Frequency bias coefficients
		R	2.4	2.7	Governor speed regulation parameters

illustrates the mathematical structure of a dual-area power system. The governor’s response, denoted by $\mathsf{\Delta }{X}_G$, indicates how the input control signal $u$ triggers the valve openings. Figure 2 depicts this action occurring within a negative feedback loop characterized by a gain of $1/R$. The governor’s actuation is affected by the LFC gains and the speed regulation parameter, $R$, with the load frequency deviation serving as the input. Consequently, the control signal, $u$, produced by the controller regulates the frequency deviation, $\mathsf{\Delta }f$, resulting from load fluctuations in each area of the power system.

2.4. Integrated AVR-LFC Power System

The relatively loose dynamic interaction between the Automatic Voltage Regulator (AVR) and Load Frequency Control (LFC) systems permits the implementation of independent control approaches for voltage and frequency in each zone. Even so, actions taken by the AVR system to regulate terminal voltage can significantly influence real power output [53]. Consequently, the AVR has a direct and substantial impact on the LFC loop. Figure 2 illustrates the AVR loop, highlighting the coupling coefficients that define the interdependencies within the system. These include the influence of small variations in stator electromotive force (emf) on real power generation (α₁), the impact of minor rotor angle deviations on terminal voltage (α₂), the effect of rotor angle changes on stator emf (α₄), and how changes in stator emf affect the rotor angle (α₃) [60]. Additionally, $K_{12}$ represents the synchronization coefficient between the two interconnected areas.

Table 3. AVR Coupling confidents and synchronization coefficient.

Area	α1	α2	α3	α4	βS	${\mathit{K}}_{12}$
1	0.3	0.1	0.5	1.4	1.5	0.55
2	0.3	0.1	0.5	1.4	1.5

presents the value of coupling coefficients as well as the synchronization coefficient. Figure 3 shows the LFC and AVR loops simply.

2.5. Mathematical Representation of the Power System

The interconnection of the two areas through tie-lines supports power sharing under normal conditions. However, discrepancies between generation and load in any area may induce frequency oscillations throughout the network. The dynamics of the LFC loop are described as follows.

$$∆f=-{\displaystyle \frac{∆f}{T_P}}+{\displaystyle \frac{K_P}{T_P}}∆P_G-{\displaystyle \frac{K_P}{T_P}}∆P_e-{\displaystyle \frac{K_P}{T_P}}∆P_{tie}-{\displaystyle \frac{K_P}{T_P}}∆P_{Load}$$

(1)

$$∆P_G=-{\displaystyle \frac{1}{T_T}}∆P_G+{\displaystyle \frac{1}{T_T}}∆X_G$$

(2)

$$∆X_G=-{\displaystyle \frac{1}{{RT}_G}}∆f-{\displaystyle \frac{1}{T_G}}∆X_G+{\displaystyle \frac{1}{T_G}}u$$

(3)

$$∆P_{tie i,j}={2\pi T}_{ij}(∆f_i-∆f_j)$$

(4)

In this context, $\mathsf{\Delta }f$, $∆P_G$, $∆X_G$, and $∆P_{tie}$ represent the deviations in frequency (Hz), generator power output (p.u.), governor valve position (p.u.), and tie-line power (p.u.), respectively. The parameters $T_P$, $T_T$, $T_G$, $R$, and $K_{\mathrm{B}}$ correspond to the time constants of the load (s), turbine (s), and governor (s), the speed regulation coefficient (Hz/p.u.), and the frequency bias factor (p.u./Hz), respectively. The Area Control Error (ACE), denoted as $e\left(t\right)$, serves as the input to the (1+PDD²)-(1+TI) controller and is formulated as a weighted sum of frequency deviation and tie-line power deviation, as expressed below:

$$e\left(t\right)=K_B∆f_i+∆P_{tie}$$

(5)

The AVR system is composed of the amplifier, exciter, generator field, and sensor as its primary components. Terminal voltage is constantly measured and evaluated against a preset reference voltage, with the resulting error signal serving as the input to the (1+PDD²)-(1+TI) controller. This control signal is then amplified and directed to the excitation system to adjust the generator field excitation. Even a small frequency variation results in a change in the rotor angle ($\delta$), and the real power output can be expressed as follows [2,61]:

$$∆P_e=P_S∆\delta +K_{1}E$$

(6)

The variation in terminal voltage is given as

$$∆V=K_2∆\delta +K_{3}E$$

(7)

The generator field transfer function is formulated as

$$E={\displaystyle \frac{K_F}{1+{sT}_F}}(E_e-K_4∆\delta )$$

(8)

3. Control Methodology and Problem Overview

In this section, the (1+PDD²)-(1+TI) controller design is detailed, with its parameters optimized using the MSO algorithm, to effectively address challenges in LFC and AVR. Earlier research indicates that system uncertainties pose challenges for traditional controllers. Leveraging the combined advantages of the (1+TI) and (1+PDD²) configurations, the proposed controller enhances the effectiveness and robustness of regulating frequency and voltage.

3.1. MSO Algorithm

3.1.1. Motivation

A mirage is a common optical phenomenon caused by specific meteorological and geographical conditions. It typically forms when sunlight heats the ground, creating a temperature gradient in the air. These gradients lead to variations in air density and, consequently, the refractive index. As light passes through these layers, it bends due to the density differences. However, since the human brain assumes light travels in straight lines, it perceives a virtual image, known as a mirage. Mirages are mainly classified into two types: superior and inferior, based on their formation and the position of the image relative to the observer.

• A superior mirage occurs when warm air lies above cooler air, reversing the normal temperature gradient. This causes light rays to bend downward, making distant objects appear above the horizon. These are often observed over cold surfaces, such as polar regions or cold seas, and metaphorically suggest seeing beyond immediate limits.
• An inferior mirage forms when hot air is near the ground and cooler air lies above it. This causes light to bend upward, creating a distorted or floating image. A common example is the shimmering illusion seen on roads during hot weather. It symbolizes seeing only a partial or misleading view of reality.

Figure 4 illustrates the path of light in a superior mirage, where the refractive index decreases with height ($n_1>n_2>n_3$). According to Fresnel’s law, total internal reflection occurs when $n_{above}$ · $sin \alpha >n_{below}$, where $n_{above}$ refers to the denser lower layer, $n_{below}$ to the upper layer, and α is the angle of incidence. Due to the gradual change in refractive index, light bends progressively, and the observer perceives the mirage at the point where the final refracted ray appears to originate from a virtual extension of the incident path.

Figure 5 depicts the trajectory of light rays associated with an inferior mirage. In such a situation, the refractive index gradient follows the order $n_1>n_2>n_3$, Total internal reflection occurs when the condition $n_{below}$ ⋅ $sin\alpha >n_{above}$ is satisfied. The stratified refractive index causes light rays to bend upward, resulting in an inverted and magnified virtual image, which observers see along the backward extension of the refracted ray.

3.1.2. Mathematical Model

This section provides a detailed explanation of MSO, based on the underlying concepts of how mirages form. It leverages the concepts of both inferior and superior mirages along with their respective advantages. The superior mirage forms the basis for an approach for global optimization, while the inferior mirage corresponds to a local optimization approach. MSO is developed by integrating these global and local search strategies.

Formulating the Starting Solution Set

The position serves as an observation point that updates itself depending on either the superior or inferior mirage. Consequently, every observation point is mapped as a vector in an n-dimensional coordinate system.

$$X=\left[\begin{array}{ccc}{X}_{11}& \dots & X_{1d}\\ ⋮& \ddots & ⋮\\ X_{p1}& \dots & X_{pd}\end{array}\right]$$

(9)

Here, $p$ indicates the highest number of sensing points, and $d$ refers to the maximum dimensionality of each observation point. Starting solutions must meet the limitations in every dimension. Therefore, to generate an individual’s vector, the formula below is applied:

$$x_i={lb}_i+r·\left({ub}_i-{lb}_i\right),i=1,2,\dots d$$

(10)

Here, $x_i$ represents the $i$-th current individual’s dimension, ${lb}_i$ is the Lower constraint on dimension $i$, ${ub}_i$ is the higher constraint of dimension $i$, and $r$ represents a randomly selected value ranging from 0 to 1.

Superior Mirage Strategy

A superior mirage enables the observation of virtual depictions of remote objects, primarily due to the reflection of light along curved paths. Based on this phenomenon, a global exploration method inspired by the formation principles of superior mirages, as illustrated in Figure 6.

To simulate this effect, a stratified refractive index line is modeled. Consider $\beta$ as the angle formed between the ground and the stratification line, and define $\alpha$ as the angle formed between the incoming line at the initial position and the perpendicular to the horizontal surface at that point. When the condition $n_{above}$ ⋅ $sin\alpha >n_{below}$ is met, total internal reflection occurs, facilitating the observer’s perception of a reflected mirage representing a different exploration objective.

Using the following angles:

A: between the direction of reflection and the horizontal reference;

B: between the incident line and the reflected line;

C: between the incident line and the boundary of the stratified refractive index;

D: between the perpendicular to the horizontal reference line and the incident line.

The position of the exploration location can be derived accordingly.

$$∆X_{lower}={\displaystyle \frac{sinB\cdot h\cdot sinC}{sinD\cdot sinA}}$$

(11)

Here, $∆X$ refers to the horizontal span between the starting location and the target point, and $h$ indicates the vertical separation between the starting point and the stratified refractive index layer. A larger value of $h$ facilitates escaping from local optima. Therefore, $h$ refers to the displacement between the current point from the best solution.

$$h_{ij}=\left|{gbest}_j-x_{ij}\right|\cdot rand+1$$

(12)

Here, $h_{ij}$ refers to the $j$-th dimensional value of the $i$-th starting point, with its lower value set to at least 1 to ensure that the update step for each individual remains reasonable. ${gbest}_j$ presents the $j$-th component of the global best solution, and rand denotes a number randomly generated within the range [0, 1]. The value of $h_{ij}$ is constrained by the bounds specified in Equation (13).

$$1\le h_{ij}\le \left[10\cdot arctanh\left.\left(1-{\displaystyle \frac{t}{t_{max}}}\right.\right)+1\right]$$

(13)

Here, $t$ represents the iteration in progress count, and $t_{max}$ is the higher limit of iterations. These parameters help ensure that $h$ varies within a substantial, interval excluding zero, maintaining the global search capability of the approach.

Since the angles $\alpha$ and $\beta$ refer to the increment size for an individual’s movement, it is crucial to satisfy the condition $\pi /2-\beta >\alpha$. This ensures that the incident light enters the stratified layer of refractive index, producing a reflected beam. In this study, the values of $\alpha$ and $\beta$ are calculated using Equation (14).

$$\left\{\begin{array}{c}\alpha =rand\cdot \left({\displaystyle \frac{\pi }{9}}\right) \\ \beta =rand\cdot \left({\displaystyle \frac{\pi }{4}}-{\displaystyle \frac{\propto }{2}}\right)\end{array}\right.$$

(14)

If $\beta$ is less than $\pi /2$, the incident ray approaches at an angle $\alpha =\beta$, measured as an angle measured to the right of the perpendicular at the stratified refractive index layer. Given that the angle formed between the incident ray and the stratification line is $\pi /2$, the resulting reflection will form a right angle with the incident line.

MSO considers three scenarios for incident ray, defined within the bounds $\alpha =\beta$ and $\alpha =\pi /2$. Since the conditions $\beta <\pi /2$ and $\beta >\pi /2$ are symmetrical, a parameter $a=\pm 1$ is used to indicate whether the current scenario is positioned on either side of the perpendicular line to the horizontal reference, left or right, simplifying the explanation and reducing the need to separately handle the $\beta >\pi /2$ case.

Case 1: If the incident light points to the left of the perpendicular to the horizontal reference line, the step size for each individual in the superior mirage search mode can be calculated by integrating Equations (11)–(14).

$$∆X_{lower}=a\cdot \left({\displaystyle \frac{sin\left(\pi -2\alpha -2\beta \right)\cdot h\cdot sin(\frac{\pi }{2}+\beta )}{sin(\alpha -\beta )\cdot sin(\alpha -2\beta )}}\right)$$

(15)

Case 2: If the incident light is directed to the right of the perpendicular to the horizontal reference line and the condition $\beta <\alpha <\pi /2$ is met, the individual incremental size in the superior mirage search mode is given by combining Equations (11) Through (14).

$$∆X_{lower}=a\cdot \left({\displaystyle \frac{sin(\pi -2\alpha +2\beta )\cdot h\cdot sin(\frac{\pi }{2}-\beta )}{sin(\pi -\alpha +\beta )\cdot sin(\alpha -2\beta )}}\right)$$

(16)

Case 3: If the incident line points to the right of the perpendicular to the horizontal reference line and $\beta <\alpha <\pi /2$ holds true, the step size of individual in the superior mirage search mode can be obtained by integrating Equations (11) through (14).

$$∆X_{lower}=a\cdot \left({\displaystyle \frac{sin(-\pi +2\alpha -2\beta )\cdot h\cdot sin(\frac{\pi }{2}-\beta )}{sin(\pi -\alpha +\beta )\cdot sin(\pi -\alpha +2\beta )}}\right)$$

(17)

Finally based on Equations (15)–(17), the formula for updating the initial position in the inferior mirage can be formulated as follows:

$$x_{ij}^{t+1}=x_{ij}^t{+∆X}_{lower}$$

(18)

Here, $t$ represents the current iteration count.

Ultimately, the superior mirage approach supplies strong exploration capabilities, primarily due to its ability to detect distant targets. This approach relies on the principle of light reflection to guide individuals in performing global updates. To implement this mechanism, MSO organizes the superior mirage into three different cases, as defined by Equations (15)–(17). Using these formulas, all individuals are capable of updating their positions globally.

Inferior Mirage Strategy

An inferior mirage enables the observation of magnified virtual images of objects, primarily caused by the refraction of light along its path. The localized false image seen in an inferior mirage result from this refractive behavior. Based on this phenomenon, a localized exploration strategy according to the manner of downward mirages, as shown in Figure 7.

The approach starts by setting a stratified reference line at a displacement $h$ from the starting point. A stratification line for the refractive index is then set up between this horizontal reference and the existing stratification line. The observation point, along with its associated reference line, is positioned at a displacement of $2h$ from the starting location. In this set up, the search process starts from the observation position along the optical line. The point where the refracted ray intersects the horizontal reference line corresponds to the actual object, While the mirage is indicated by the point where the extended incident ray intersects that same line. This configuration demonstrates that inferior mirages occur at closer ranges and with larger angles, making them highly effective for local optimization. According to Equations (19) and (20), both the magnitude of $h$ and the direction from the present solution to the optimal solution are defined and limited.

Case 1: When the individual under consideration is not the population’s best performer.

$$\left\{\begin{array}{c}{h}_{ij}=\left|{gbest}_j-x_{ij}\right|\cdot rand \\ D={\displaystyle \frac{({gbest}_j-x_{ij})\cdot rand}{h_{ij}}}\end{array}\right.$$

(19)

Case 2: When the individual under consideration is the population’s best performer.

$$\left\{\begin{array}{c}{h}_{ij}=\left|\pm 0.05\cdot rand\right| \\ D={\displaystyle \frac{\pm 0.05\cdot rand}{h_{ij}}}\end{array}\right.$$

(20)

Here, $D$ takes a value of $\pm 1$, where $+1$ indicates that the optimal solution lies toward the upper bound relative to the current individual, and $-1$ indicates that it lies toward the lower bound. $\gamma$ indicates the angular difference between the stratification line and the stratified reference line, varying from 0 to $\pi /2$, and it is initialized based on this range.

$$\gamma ={\displaystyle \frac{\pi (t_{max}-t)}{2t_{max}}}\cdot rand$$

(21)

The angle $\phi$, which is determined by the angle between the incoming ray and the perpendicular at the observation site, is bounded by two constraints. The lower bound, known as the angle of entrapment, corresponds to the case where the refracted light just reaches the starting position. The upper bound is defined by the point at which the refracted ray intersects the point where the refractive index stratification line meets the horizontal reference line. The value of $\phi$ is initialized according to Equation (22).

$$\phi =\left(arctan\left({\displaystyle \frac{1}{2tan\gamma }}\right)-arctan\left({\displaystyle \frac{sin\gamma \cdot cos\gamma }{1+{\mathit{sin}}^2\gamma }}\right)\right)\cdot rand+arctan\left({\displaystyle \frac{sin\gamma \cdot cos\gamma }{1+{\mathit{sin}}^2\gamma }}\right)$$

(22)

Based on Fresnel’s reflection principle, the angle of incidence and the refractive indices on either side of the refractive index stratification line are governed by the refraction angle $\omega$, which is the angle formed between the incident light and the normal to the stratification line.

$$\omega =arcsin\left({\displaystyle \frac{n_2}{n_1}}\cdot sine(\phi +\gamma )\right)$$

(23)

Here, $n_1$ represents the index of refraction below the interface of stratification, and $n_2$ represents the refractive index above it, with the condition that $n_1<n_2$. Using Equations (19) through (23), the current individual’s movement can then be evaluated.

$$∆X_{upper}={\displaystyle \frac{h}{tan\gamma }}-{\displaystyle \frac{\left(\frac{h}{sin\gamma }-\frac{hsin\phi }{cos(\phi +\gamma )}\right)\cdot cos\omega }{cos(\omega -\gamma )}}$$

(24)

Ultimately, by combining Equations (19), (20), and (24), the update equation for the initial position in the inferior mirage strategy can be formulated as follows:

$$x_{ij}^{t+1}=x_{ij}^t+D∆X_{upper}$$

(25)

In conclusion, the inferior mirage approach provides strong exploitation capabilities, due to the characteristics of inferior mirages that enable the observation of enlarged virtual images of objects. This approach leverages the principle of ray refraction to direct individuals in conducting localized searches. MSO carries out the inferior mirage search mechanism via Equation (24), enabling all members to carry out effective local exploitation based on this formulation.

Equilibrium Between Exploration and Exploitation

By utilizing a strategy where a subset of individuals is dedicated to exploration while all individuals participate in exploitation, the model ensures comprehensive coverage of the search space along with effective convergence to the optimal solution. At the outset, all individuals are ranked, with the current best-performing individual placed at the top. Subsequently, $n$ individuals are randomly chosen from the remaining population (excluding the top-ranked individual) based on Equation (26).

$$n=ceil\left(20\cdot {\displaystyle \frac{t_{max}-t}{t_{max}}}\right)$$

(26)

Here, $ceil$ refers to the ceiling function, which rounds a number up to the nearest integer. The value of $n$ is constrained to be no less than 1 and no more than 20.

Once the superior and inferior mirage search processes are completed, the individuals are re-ranked. In MSO, the leading individual always represents the current best solution. The superior mirage strategy maintains this optimal solution by excluding it from updates and focusing on a selected group of individuals. In contrast, the inferior mirage strategy involves the entire population in the search process.

3.1.3. Execution of the MSO Algorithm

This subsection outlines the MSO algorithm’s execution process. MSO begins by initializing the solution set and sorting all solutions from best to worst fitness values to ensure proper algorithm operation. The algorithm employs two complementary strategies. The upper mirage strategy randomly selects candidate solutions (excluding the optimal solution) for updating, providing strong global search capability. Its implementation decreases with iteration progress to enable subsequent local search. This strategy updates candidate solutions using Equations (15) to (17). The lower mirage strategy updates all candidate solutions with strong local search capability, using Equations (19) and (20) for solution modifications. After completing the lower mirage strategy, the current population is ranked by fitness values and optimal individuals are selected for retention. The complete MSO execution procedure is clearly illustrated in Figure 8.

3.1.4. Computational Complexity

Computational complexity is a key metric for evaluating an algorithm’s efficiency. This section analyzes both the time and space complexity of the Mirage Search Optimization (MSO) algorithm.

(a)

Time complexity

The Mirage Search Optimization (MSO) algorithm begins by initializing a population to generate an initial set of candidate solutions for the optimization process. The time complexity of this initialization phase depends on the problem’s dimensionality and the population size. In this study, the dimension size ($d$) is 48, and the population size ($popsize$) is 30. Therefore, the time complexity for population initialization is $O(48\times 30)=O \left(1440\right)$.

This algorithm incorporates two strategies: the superior mirage strategy and the inferior mirage strategy. The superior mirage strategy updates a subset of $n$ individuals according to Equation (18), with a time complexity of $O(t_{max}\times d\times n)$, where $t_{max}$ is the maximum number of iterations (200). Similarly, the inferior mirage strategy has a time complexity of $O(t_{max}\times d\times popsize)$.

Thus, the overall runtime time complexity of the proposed MSO algorithm is as follows:

$$O\left(t_{max}\times d\times \left(popsize+n\right)\right)=O\left(200\times 48\times \left(30+n\right)\right)$$

(27)

This expression highlights that the computational cost scales linearly with the number of iterations, problem dimensionality, and population size.

(b)

Space complexity

The space complexity of the Mirage Search Optimization (MSO) algorithm refers to the maximum memory required during its execution. It primarily depends on the dimensionality of the problem ($d$) and the population size ($popsize$). Since MSO must store and continuously update candidate solutions throughout the optimization process, the space complexity can be expressed as $O(d\times popsize)$.

3.2. The Precise Design of the Proposed (1+PDD²)-(1+TI) Regulator

This section presents the structure of the proposed (1+PDD²)-(1+TI) controller. The controller includes two main elements: the first is the (1+TI) controller, known for its superior efficiency compared to both traditional and modern controllers, and for its fast response to system fluctuations. However, the (1+TI) controller is limited by its failure to enhance both phase margin and the system’s equilibrium accuracy. To address this issue, the (1+PDD²) controller is typically employed. Figure 9 illustrates the architecture of the (1+PDD²)-(1+TI) controller, and its mathematical representation is given by the following equations.

$$H\left(s\right)=C1\left(s\right) \cdot C2\left(s\right)$$

(28)

$$C_1\left(s\right)=1+K_T{·S}^{-\left({\displaystyle \frac{1}{N_T}}\right)}+K_I·{\displaystyle \frac{1}{S}}$$

(29)

$$C_2\left(s\right)=1+K_P+K_D·\left({\displaystyle \frac{N_D· S}{S+N_D}}\right)+K_D·\left({\displaystyle \frac{N_D· S}{S+N_D}}\right)·K_{DD}·\left({\displaystyle \frac{N_{DD}· S}{S+N_{DD}}}\right)$$

(30)

Ultimately, the optimal values of K_P, K_D, K_DD, N_D, N_DD, K_T, N, K_I for (1+PDD²)-(1+TI) controllers are determined using the MSO algorithm, aiming to minimize the objective function defined in Equation (31).

$$J=ITSE={\int }_0^t{t\left[\right(∆f_1)}^2{+{(∆f_2)}^2+(∆P_{tie})}^2+{(∆V_1)}^2{+(∆V_2)}^2]dt$$

(31)

These coefficient ranges are specified in two arrays within the algorithm’s code, defined as the low limit array and the high limit array, representing the minimum and maximum values, respectively, as presented in Equation (32). The lower bounds (min value) are defined as lb = [0 0 0 100 100 0 1 0], and the upper bounds (max value) as ub = [5 5 0.1 500 500 5 10 5].

$$\left\{\begin{array}{c}\begin{array}{c}{K}_{P}min<K_P<K_{P}max\\ K_{D}min<K_D<K_{D}max\\ K_{DD}min<K_{DD}<K_{DD}max\end{array}\\ \begin{array}{c}{N}_{D}min<N_D<N_{D}max\\ N_{DD}min<N_{DD}<N_{DD}max\\ K_{T}min<K_T<K_{T}max\end{array}\\ \begin{array}{c}Nmin<N<Nmax\\ K_{I}min<K_I<K_{I}max\end{array}\end{array}\right.$$

(32)

Therefore, this study seeks to decrease (ITSE) using the MSO algorithm to identify the optimal coefficients of the proposed controller and to subsequently evaluate its performance against that of other controllers with respect to maximum overshoot (MO), maximum undershoot (MU), settling time (ST), and rise time (RT).

Figure 9. Proposed controller structure.

Figure 9. Proposed controller structure.

4. Simulation Results and Analysis

This section assesses the behavior of a two-area interconnected system regulated by the proposed (1+PDD²)-(1+TI) controller, based MSO algorithm. The evaluation is conducted in two main steps:

The MSO method is first benchmarked against several widely recognized techniques from the literature in terms of performance and efficiency, including ChOA, DOA, PSO, GTO, and GBO. These methods are benchmarked using 200 iterations and a population size of 30, under a sudden load rising at t = 0 specifically, a 2% increase in Area 1 and a 1.5% increase in Area 2. The MSO algorithm outperforms the other techniques in terms of fitness, with improvements of approximately 44%.

Second, the robustness and stability of the proposed (1+PDD²)-(1+TI) controller, optimized by MSO, are evaluated against other MSO-tuned controllers, including TID, FOPID, FOPI-PIDD², (1+PD)-PID, and PID controller. This comparison is proceeded under various operating conditions, such as sudden load disturbances at t = 0, random load fluctuations, time-varying reference voltages in both areas, and system parameter variations. The evaluation focuses on each controller’s effectiveness in controlling frequency, voltage, and tie-line power deviation (ΔPtie) of the system. These two evaluation steps are discussed extensively in the upcoming subsections.

4.1. Evaluating the Effectiveness of the MSO Algorithm

The convergence behavior of the six optimization algorithms (ChOA, DOA, PSO, GTO, GBO, and the proposed MSO) is illustrated in Figure 10. This figure presents the best cost (objective function value) versus iteration number across 200 iterations. The convergence curves provide insight into the exploration and exploitation capabilities of each algorithm and their efficiency in reaching optimal solutions. From the figure, it is evident that the MSO algorithm not only achieves the lowest final fitness value but also converges more rapidly than the other techniques. During the initial stages (first 50–80 iterations), MSO shows a steep decline in the fitness value, indicating a strong global search ability. As iterations progress, MSO continues to refine the solution with stable convergence, avoiding premature stagnation or divergence common issues observed in some of the other algorithms. In contrast, algorithms like ChOA and DOA exhibit slower convergence and tend to plateau early, suggesting weaker exploitation ability and less efficiency in navigating the search space. PSO and GTO show moderate performance but lack the sharp convergence rate of MSO. GBO performs relatively better among the compared methods but still falls short of MSO’s final solution quality and convergence speed. This behavior validates the superior adaptability and robustness of MSO in fine-tuning the (1+PDD²)-(1+TI) controller’s parameters. It also highlights MSO’s capability to balance exploration and exploitation phases effectively, making it particularly suitable for complex, nonlinear control problems such as integrated LFC-AVR systems.

Table 4. Optimum settings of the proposed (1+PDD²)-(1+TI) controller tuned by the six optimization algorithms.

	Parameters	ChOA	DOA	GTO	PSO	GBO	MSO
LFC1	KP	0	3.15	4.99	5	4.48	4.49
	KD	5	5	4.99	5	4.87	2.05
	KDD	7.95 × 10−5	0.1	0.02	0	0.025	0.05
	ND	142.93	202.02	500	500	135.11	351.78
	NDD	370.18	223.71	100	100	481.46	204.23
	KT	0.02	2.65	5	0	2.31	4.024
	NT	1	10	1.012	10	1.30	1.91
	KI	5	3.46	2.62	5	4.99	3.15
AVR1	KP	0	4.96	2.25	4.99	0.52	4.62
	KD	2.16	4.23	0.74	0.99	0.37	1.15
	KDD	42.14 × 10−5	0.03	93.06 × 10−4	0.02	21.2 × 10−4	40.78 × 10−5
	ND	423.36	494.57	100.65	500	364.87	481.22
	NDD	100	500	500	500	496.52	190.67
	KT	0.12	4.80	1.87	5	4.49	0.74
	NT	1.08	1.37	9.88	10	9.87	9.96
	KI	5	2.15	0.29	0	0.57	0.17
LFC2	KP	96.96 × 10−5	5	4.99	5	4.35	4.02
	KD	5	4.96	4.99	5	2.48	4.32
	KDD	89.66 × 10−4	0.1	98.81 × 10−3	57.35 × 10−4	12.37 × 10−5	0.05
	ND	206.28	456.97	100.02	100	313.58	251.57
	NDD	253.15	196.01	100	112.78	133.39	344.77
	KT	58.83 × 10−4	5	4.99	1.93	4.99	0.58
	NT	1.21	9.53	1.25	10	9.85	3.76
	KI	4.73	3.39	5	5	4.97	4.81
AVR2	KP	0	0	2.07	5	1.40	4.92
	KD	0.79	1.80	0.71	1.03	0.55	1.23
	KDD	0	0	18.92 × 10−4	13.77 × 10−3	16.25 × 10−4	16.96 × 10−4
	ND	139.51	500	148.27	500	496.31	354.09
	NDD	500	107.50	317.20	500	124.01	465.33
	KT	2.55	0	1.39	5	3.13	0.64
	NT	1.65	2.98	10	10	9.99	7.28
	KI	19.02 × 10−5	5	0.35	0.06	0.36	0.16
Fitness	ITSE	0.044	0.05	0.03	0.033	0.03	0.028

summarizes the optimal controller parameters obtained through each optimization algorithm for the proposed (1+PDD²)-(1+TI) controller.

The performance of the MSO algorithm is evaluated through a comparative analysis with five other metaheuristic algorithms: ChOA, DOA, GTO, PSO, and GBO. Each algorithm is executed for 30 independent runs per test function. The population size is set to 40, with a maximum of 300 iterations. The optimal fitness values obtained are summarized in

Table 5. The best fitness functions achieved by the six strategies.

Test Functions		ChOA	DOA	GTO	PSO	GBO	MSO
$\mathit{F}\mathit{n}.\mathbf{1}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^D}{z}_i^2$	$2.48\times {10}^{-37}$	$6.93\times {10}^{-103}$	$3.02\times {10}^{-105}$	$7.15\times {10}^{-131}$	$7.95\times {10}^{-152}$	0
$\mathit{F}\mathit{n}.\mathbf{2}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^D}\left|z_i\right|+{\displaystyle \prod _{i=1}^D}\left|z_i\right|$	$1.01\times {10}^{-21}$	$9.52\times {10}^{-53}$	$7.81\times {10}^{-55}$	$7.44\times {10}^{-74}$	$2.40\times {10}^{-80}$	$1.73\times {10}^{-155}$
$\mathit{F}\mathit{n}.\mathbf{3}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^D}{\left({\displaystyle \sum _{j=1}^D}{z}_j\right)}^2$	$1.17\times {10}^{-25}$	$8.49\times {10}^{-28}$	$2.64\times {10}^{-99}$	$6.76\times {10}^{-102}$	$2.96\times {10}^{-105}$	$5.03\times {10}^{-118}$
$\mathit{F}\mathit{n}.\mathbf{4}\left(\mathit{z}\right)$	$\begin{array}{c}\\ \underset{i}{\mathit{max}}\end{array}\left\{\left|z_i\right|, 1\le i\le D\right\}$	$2.09\times {10}^{-16}$	$7.90\times {10}^{-50}$	$2.42\times {10}^{-54}$	$9.43\times {10}^{-66}$	$2.37\times {10}^{-68}$	$3.04\times {10}^{-170}$
$\mathit{F}\mathit{n}.\mathbf{5}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^{D-1}}\left[100{\left(z_{i+1}-z_i^2\right)}^2+{\left(z_i-1\right)}^2\right]$	9.63	7.85	6.78	2.76	2.16	0
$\mathit{F}\mathit{n}.\mathbf{6}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^D}{\left(\left|z_i+0.5\right|\right)}^2$	$4.86\times {10}^{-27}$	0	0	0	0	0
$\mathit{F}\mathit{n}.\mathbf{7}\left(\mathit{z}\right)$	${\displaystyle \sum _{i=1}^D}{iz}_i^4+rand\left(0,1\right)$	$9.64\times {10}^{-4}$	$4.48\times {10}^{-4}$	$2.50\times {10}^{-5}$	$2.42\times {10}^{-5}$	$1.89\times {10}^{-4}$	$1.41\times {10}^{-4}$
Total consumption time for all test function		0.6091161	0.6109206	0.5104028	0.7284502	0.5891876	0.5266098

, while Figure 11 illustrates the convergence curves for all algorithms across the benchmark functions. The results clearly demonstrate the superior performance of MSO compared to the other methods.

4.2. Robustness Analysis of the Proposed (1+PDD²)-(1+TI) Controller

For evaluating the performance of the proposed controller in improving the performance of the dual-area interconnected power system, a series of simulations were executed using the MATLAB/Simulink environment. Simulations were designed to evaluate performance across multiple operating conditions and challenges. The results are presented in the following subsections, each corresponding to a specific scenario:

• Scenario I: A sudden change in demanded load is introduced at t = 0 s, with a 2% increase in Area 1 and a 1.5% increase in Area 2;
• Scenario II: Random load disturbances are applied to the system to assess its dynamic response under unpredictable conditions;
• Scenario III: The performance of controller is evaluated in the presence of typical system nonlinearities, such as Generation Rate Constraint (GRC) and Governor Dead Band (GDB);
• Scenario IV: A time-varying reference voltage is provided to examine how well the controller tracks dynamic reference signals;
• Scenario V: A sensitivity analysis is proceeded to assess the system’s strength against system parameters changes.

4.2.1. Scenario I: 2% Load Increase in Area 1 and 1.5% Load Increase in Area 2

This subsection evaluates the system’s performance under a sudden load increase—2% in Area 1 and 1.5% in Area 2 at $t=0$s. The goal is to assess the ability of the system to maintain frequency and voltage stability during such disturbances. Simulation results show that the proposed (1+PDD²)-(1+TI) controller, optimized by the MSO algorithm, delivers superior performance in both LFC and AVR functions.

Compared to MSO-tuned TID, FOPID, FOPI-PIDD², (1+PD)-PID, and conventional PID controllers, the proposed controller exhibits significantly reduced undershoot and overshoot. Figure 12 and Figure 13 illustrate frequency and voltage responses, confirming the controller’s robustness and improved dynamic stability. Specifically, Figure 12a,b show frequency deviations (ΔF1, ΔF2), while Figure 13a,b show terminal voltage variations (ΔV1, ΔV2). Figure 14 presents the improved tie-line power flow response (ΔPtie).

Quantitatively, the proposed controller achieves the lowest undershoot values of −93 × 10⁻³ Hz (ΔF1) and −48.3 × 10⁻³ Hz (ΔF2), with zero overshoot. Voltage responses show minimum undershoots of −0.15 V and −0.11 V, and overshoots of only 0.04 V. For tie-line power, the lowest deviations are −11.8 × 10⁻⁴ PU and 21.5 × 10⁻⁴ PU.

In terms of fitness function, the proposed controller outperforms (1+PD)-PID, FOPI-PIDD², TID, FOPID, and PID controllers by 44%, 64%, 82%, 85%, and 86.6%, respectively, achieving a minimum ITSE value of 0.028. These results confirm the effectiveness and reliability of the (1+PDD²)-(1+TI) controller for dual LFC-AVR control. Full numerical results are provided in

Table 6. Comprehensive statistical analysis of important performance metrics.

Controller	ΔF1			ΔF2			ΔPtie			ΔV1		ΔV2		ITSE
	MOS	MUS	ST	MOS	MUS	ST	MOS $(\ast {10}^{-4}$)	MUS $(\ast {10}^{-4}$)	ST	MOS	ST	MOS	ST
TID	0.034	−0.596	8	0.045	−0.286	8	1.2	−5.2	10	1.14	3.7	1.14	3.7	0.104
FOPID	0	−0.295	6	0	−0.369	6.5	7.6	−3	5	1.08	4	1.08	3.9	0.038
FOPI-PIDD2	0	−0.14	10	0	−0.137	10	0	−4.6	8	1.05	3.8	1.06	3.9	0.031
(1+PD)-PID	0	−0.093	4	0	−0.089	3	0.67	−4.6	6	1.07	3.5	1.02	2.5	0.031
PID	0	−0.358	5	0	−0.349	5	6.8	−2.17	4.5	1.2	3	1.2	2.7	0.029
(1+PDD2)-(1+TI) (proposed)	0	−0.093	3	0	−0.048	2.22	2.15	−1.18	4	1.04	2.2	1.4	2.2	0.028

.

Figure 15 presents the convergence profiles of six different controllers optimized using the Mirage Search Optimization (MSO) algorithm, where the fitness value is plotted on a logarithmic scale against the number of iterations. The plot provides valuable insights into how quickly and effectively each controller structure converges to a near-optimal solution when tuned using the same optimization strategy. From the figure, the proposed (1+PDD²)-(1+TI) controller exhibits the fastest and smoothest convergence behavior, reaching a final fitness value close to 10⁻², significantly lower than the others. It achieves this within the first 50 iterations, with minimal fluctuation in subsequent iterations, demonstrating both fast convergence and solution stability. This confirms the structure’s strong synergy with the MSO algorithm and its efficiency in handling the complex dynamics of LFC-AVR systems. In contrast, controllers like FOPI–PIDD² and TID show slower and less stable convergence, with FOPI–PIDD² having large initial fluctuations and a delayed drop-in fitness before stabilizing around iteration 100. The TID controller remains trapped in a higher fitness region (around 10⁻¹), indicating limited adaptability or expressiveness in addressing the multi-objective tuning needs of the system. The FOPID, PID, and (1+PD)-PID controllers converge moderately well but plateau at higher fitness values compared to the proposed controller, suggesting weaker optimization potential despite using the same tuning algorithm. Overall, the results highlight that controller structure has a significant impact on the convergence behavior, even under identical optimization settings. The proposed hybrid structure not only accelerates convergence but also enables MSO to find a more optimal solution, validating its effectiveness in regulating frequency, voltage, and tie-line power within interconnected power systems.

Figure 16 presents the fitness vs. time plot. It also reaches optimal performance in significantly less computational time, demonstrating not only accuracy but also computational efficiency. Other controllers, such as FOPI–PIDD² and TID, exhibit slower convergence, larger fluctuations, and longer computation times. FOPID, PID, and (1+PD)-PID show moderate performance but plateau early with higher fitness values. These comparisons confirm that controller structure greatly influences both convergence speed and resource efficiency, with the proposed hybrid structure showing clear superiority.

4.2.2. Scenario II: Random Load Variation in Both Areas

This subsection analyzes the system’s performance under random load disturbances to evaluate its dynamic response in unpredictable scenarios, as shown in Figure 17. The response curves of the six tested controllers are presented in Figure 18, Figure 19 and Figure 20, highlighting the best performance of the proposed controller utilizing MSO algorithm. For the combined LFC-AVR system, the analysis focused on minimizing overshoot (6.5 × 10⁻³ and 1.1 × 10⁻³), undershoot (−6.9 × 10⁻³ and −2.1 × 10⁻³), and settling time (0.22 s and 0.14 s) for frequency deviations (ΔF1 and ΔF2), respectively, as shown in Figure 18. Similarly, in Figure 20, the ΔPtie response indicates that the suggested controller provides the fastest return to steady state during sudden load disturbances, showing the shortest settling time (approximately 2 s) during system abnormalities. The results confirm that the (1+PDD²)-(1+TI) controller significantly outperforms others in reducing overshoot and undershoot. Moreover, terminal voltage deviations remained minimally affected by load disturbances. The recommended (1+PDD²)-(1+TI) regulator also demonstrated its capability to stabilize the system and effectively reduce the fluctuations in the power network. Upon further analysis, it was observed that the (1+PDD²)-(1+TI) controller delivered superior suppression of frequency deviation overshoots and undershoots compared to other designs. Consequently, the intelligent (1+PDD²)-(1+TI) regulator, optimized via the MSO technique, achieved performance improvements of 64% over the MSO-based FOPID, 62% over the MSO-based TID, and 60% over the MSO-based FOPI-PIDD² controllers.

Table 7. Dynamic specifications of the investigated system represented as ITSE value using different controllers under Scenario II impact.

Controller	ITSE				Total ITSE
	ΔF1	ΔF2	ΔV1	ΔV2
TID	0.1152	0.0876	0.0729	0.0514	0.3271
FOPID	0.0444	0.0368	0.0296	0.0213	0.1321
FOPI-PIDD2	0.02953	0.02038	0.01726	0.01145	0.07862
(1+PD)-PID	0.02973	0.02264	0.01678	0.01052	0.07967
PID	0.02211	0.01584	0.01297	0.00831	0.05923
(1+PDD2)-(1+TI) (proposed)	0.01460	0.01114	0.00985	0.00572	0.04131

presents the dynamic performance of the investigated system under Scenario II by comparing the Integral of Time-weighted Squared Error (ITSE) values for various controllers. The table includes the ITSE contributions from frequency deviations in both areas and voltage deviations at terminal buses. It also summarizes the total ITSE for each controller, offering a clear comparison of their effectiveness in minimizing dynamic errors. This comparison highlights the performance advantage of advanced controller structures over conventional methods.

4.2.3. Scenario III: Impact of Typical System Nonlinearities

In this subsection, Power system frequency stability is examined under the influence of common nonlinearities, namely Generation Rate Constraint (GRC) and Governor Dead Band (GDB), as illustrated in Figure 21. For the simulation, the GRC and GDB are set to ±3%/min and 0.036 Hz, respectively. These nonlinearities significantly affect system performance, resulting in noticeable overshoots (0.245 Hz and 0.097 Hz) and undershoots (rising from −0.092 to −0.417 Hz for ΔF1 and from −0.089 to −0.330 Hz for ΔF2), as shown in Figure 22. Consequently, the capability of the system to maintain frequency stability is impaired, and the settling time is adversely impacted. Moreover, terminal voltage deviations remained minimally affected by nonlinearities.

Table 8. Comprehensive statistical analysis of affected performance metrics with GRC and GRB.

(1+PDD2)-(1+TI) (Proposed Controller)	ΔF1			ΔF2			ΔPtie			ITSE
	MOS	MUS	ST	MOS	MUS	ST	MOS $(\ast {10}^{-4}$)	MUS $(\ast {10}^{-4}$)	ST
Without GRC and GRB	0	−0.093	3	0	−0.089	2.22	2.15	−1.18	4	0.028
With GRC and GRB	0.241	−0.417	5	0	−0.331	6.5	4.1	−3.6	5	0.038

presents the impact of nonlinearities, specifically Generation Rate Constraint (GRC) and Governor Dead Band (GRB), on key system responses including ΔF1, ΔF2, ΔPtie, and the fitness value (ITSE). The results show that these nonlinearities cause noticeable deviations in frequency and tie-line power. However, terminal voltage responses remain largely unaffected.

The proposed controller remains effective in minimizing the negative impacts of GRC and GDB, even under challenging conditions, ensuring reliable frequency stabilization. Its robustness and adaptive design enable it to manage these nonlinearities and maintain frequency within acceptable operational bounds. This demonstrates the controller’s strong potential for dependable frequency regulation in real-world scenarios involving typical nonlinear disturbances. Based on the performance analysis as shown in Figure 23a,b, the (1+PDD²)-(1+TI) regulator, optimized using the MSO algorithm, outperformed its counterparts with notable improvements: 6% over the MSO-based (1+PD)-PID, 10% over the MSO-based FOPI-PIDD², 12% over the MSO-based FOPID, 43% over the MSO-based TID, and 45% over the MSO-based PID controllers.

4.2.4. Scenario IV: Time-Varying Desired Output Voltage

This challenging scenario evaluates the robustness of six comparative controllers by subjecting them to time-varying reference output voltages in Area 1 and 2, as depicted in Figure 24. The response curves of the six controllers are illustrated in Figure 25, Figure 26 and Figure 27, demonstrating the highest efficiency of the proposed controller optimized via the MSO algorithm. For the integrated LFC-AVR system, the analysis focuses on minimizing overshoot (8.6 × 10⁻³ for both ΔF1 and ΔF2), undershoot (−15 × 10⁻³ for ΔF1 and −2.1 × 10⁻³ for ΔF2), and settling time (1 s for ΔF1 and 1.5 s for ΔF2), as shown in Figure 24. Likewise, Figure 27 illustrates that the terminal voltage responses (ΔV1 and ΔV2) closely follow the desired voltage trajectory with minimal overshoot (0.013 V and 0.02 V) and undershoot (−0.04 V and −0.02 V), with the proposed controller exhibiting the fastest convergence to steady state, achieving a settling time of approximately 2 s.

Regarding power exchange dynamics, the ΔPtie response (Figure 26) reveals that the proposed controller yields the lowest overshoot (1.2 × 10⁻³ PU) and undershoot (−3.07 × 10⁻³ PU) during voltage variations. The (1+PDD²)-(1+TI) regulator, tuned using the MSO algorithm, effectively suppresses frequency oscillations and enhances control performance, even under rapid changes in terminal voltage. Performance improvements of the proposed MSO-tuned (1+PDD²)-(1+TI) controller were quantified as follows: 81% over MSO-based TID, 76% over MSO-based PID, 59% over MSO-based FOPID, 20% over MSO-based FOPI-PIDD², and 30% over MSO-based (1+PD)-PID controllers.

Table 9. Dynamic specifications of the investigated system represented as ITSE value using different controllers under Scenario IV impact.

Controller	ITSE				Total ITSE
	ΔF1	ΔF2	ΔV1	ΔV2
TID	0.0976	0.0792	0.0615	0.0478	0.2861
FOPID	0.0422	0.0339	0.0258	0.0176	0.1195
FOPI-PIDD2	0.02800	0.01877	0.01628	0.01136	0.07441
(1+PD)-PID	0.02740	0.01925	0.01567	0.01082	0.07314
PID	0.02044	0.01533	0.01047	0.00812	0.05436
(1+PDD2)-(1+TI) (proposed)	0.01214	0.01041	0.00893	0.00684	0.03832

presents the dynamic performance of the investigated system under Scenario IV in terms of the Integral of Time-weighted Squared Error (ITSE) for frequency and voltage deviations across two areas. Several controllers are compared, including TID, FOPID, FOPI-PIDD², (1+PD)-PID, PID, and the proposed (1+PDD²)-(1–TI) controller. Among all, the proposed controller demonstrates the best performance with the lowest total ITSE value of 0.03832, indicating superior dynamic response and improved system stability compared to conventional and advanced control strategies.

4.2.5. Scenario V: Sensitivity Analysis of the System Under Parametric Variations

Sensitivity analysis evaluates the system’s capability to sustain performance and stability despite variations in its parameters within a defined range. In this section, the robustness of the system is examined by systematically varying key parameters of the system K_G, K_T, K_P, K_A, K_E, K_F, T_G, T_T, T_P, T_A, T_E, and T_F by $\pm 20\%$ and $\pm 40\%$ from their nominal values. This analysis is conducted while maintaining the load disturbance profile defined in Scenario I for the proposed (1+PDD²)-(1+TI) controller.

The evaluation is carried out in two steps. First, the Load Frequency Control (LFC) parameters in both Area 1 and Area 2 are varied.

Table 10. The dynamic characteristics of the (1+PDD²)-(1+TI) regulator-based system under LFC parametric variations.

Parameter	Variation	ΔF1		ΔF2		ΔPtie
		MUS (Hz)	ST (s)	MUS (Hz)	ST (s)	MOS $(\ast {10}^{-4}$) (PU)	MUS $(\ast {10}^{-4}$) (PU)
Nominal	0%	−0.093	3.8	−0.048	2.22	2.15	−1.18
KG	+40% +20%	−0.083 −0.088	4.1 4	−0.070 −0.074	4.2 4	1.89 1.94	−1.92 −1.74
	−20% −40%	−0.122 −0.125	4.4 4.6	−0.111 −0.116	4 4.1	2.7 2.9	−1.94 −1.98
TG.1	+40% +20%	−0.120 −0.115	4 4	−0.097 −0.092	4.3 4	2.8 2.61	−2.98 −2.85
	−20% −40%	−0.098 −0.095	4 4.2	−0.088 −0.084	4 4.2	2.16 2.19	−1.47 −1.59
KT	+40% +20%	−0.085 −0.088	4 4	−0.070 −0.076	3.7 3.5	1.69 1.73	−2.02 −1.94
	−20% −40%	−0.115 −0.119	4.4 4.7	−0.109 −0.113	4.5 4.6	2.61 2.72	−1.51 −1.63
TT.1	+40% +20%	−0.116 −0.106	4.1 4	−0.093 −0.091	4.4 4.1	2.37 2.29	−2.21 −2.12
	−20% −40%	−0.079 −0.070	4 4	−0.087 −0.086	4 4.5	2 1.94	−0.44 −0.31
KP.1	+40% +20%	0.084 −0.09	4 4	0.091 −0.089	4.5 4	2.23 2.18	−1.36 −1.31
	−20% −40%	−0.094 0.098	4 4.1	−0.088 −0.083	4 4.1	2.1 2.08	−0.97 −0.80
TP.1	+40% +20%	−0.095 −0.093	4.6 4.4	−0.084 −0.088	4.2 4	2.09 2.11	−0.96 −1.01
	−20% −40%	−0.089 −0.086	4 4.1	−0.09 −0.094	4.4 4.5	2.17 2.21	−1.33 −1.51
TG.2	+40% +20%	−0.097 −0.093	4.3 4.1	−0.099 −0.094	4.2 4.1	2.01 2.06	−0.77 −0.81
	−20% −40%	−0.091 −0.088	4.4 4.7	−0.084 −0.079	3.9 4.2	2.24 2.33	−1.57 −1.69
TT.2	+40% +20%	−0.095 −0.093	4 4	−0.16 −0.1	4.3 4.1	2.56 2.44	−0.49 −0.65
	−20% −40%	−0.091 −0.088	4 4.1	−0.078 −0.071	4 4	2.16 2.18	−1.95 −2.27
KP.2	+40% +20%	−0.096 −0.093	4.4 4	−0.086 −0.088	4.2 4	2.02 2.08	−0.78 −0.97
	−20% −40%	−0.092 −0.094	4 4.2	−0.09 −0.093	4 4	2.25 2.36	−1.48 −1.68
TP.2	+40% +20%	−0.096 −0.092	4.2 4	−0.091 −0.089	4.1 4	2.40 2.23	−1.63 −1.42
	−20% −40%	−0.092 −0.097	4 4.1	−0.088 −0.086	4 4.4	2.06 1.98	−0.95 −0.89

presents the performance results following Scenario I condition, under both nominal and perturbed conditions. The results indicate that the system’s dynamic responses ΔF1 (maximum overshoot), ΔF2 (maximum overshoot), ΔPtie (settling time), and the terminal voltage remain stable and largely unaffected by these variations.

In the second step, the Automatic Voltage Regulator (AVR) parameters in both areas are varied.

Table 11. The dynamic characteristics of the (1+PDD²)-(1+TI) regulator-based system under AVR parametric variations.

Parameter	Variation	ΔF1	ΔF2	ΔPtie		ΔV1			ΔV2
		ST (s)	ST (s)	MOS $(\ast {10}^{-4}$) (PU)	MOS $(\ast {10}^{-4}$	MOS (PU)	MUS (PU)	ST (s)	MOS (PU)	MUS (PU)	ST (s)
Nominal	0%	3	2.22	2.15	−1.18	1.04	0.85	2.2	1.04	0.89	2.2
KA	+40% +20%	4.8 4.5	7 5	2.30 2.21	−1.32 −1.24	1.10 1.08	0.78 0.81	3.2 2.9	1.10 1.08	0.73 0.81	3.4 2.9
	−20% −40%	4.6 5	4 5	2.04 1.91	−1.07 −1.01	1.06 1.05	0.88 0.93	2 1.8	1.07 1.08	0.87 0.94	2 1.7
TA	+40% +20%	4.9 4.5	6.8 4.5	2.31 2.21	−1.11 −1.15	1.3 1.1	0.76 0.81	3.3 2.9	1.19 1.1	0.80 0.82	3.3 2.9
	−20% −40%	4.2 4.4	3.9 4.5	2.06 1.93	−1.17 −1.15	1.4 1.7	0.86 0.88	2 1.8	1.3 1.42	0.86 0.90	1.7 1.5
KE	+40% +20%	4.8 4.4	4.7 4.1	2.32 2.21	−1.29 −1.21	1.11 1.08	0.74 0.81	3.2 2.9	1.12 1.08	0.73 0.81	3.4 2.9
	−20% −40%	4.1 4.3	4 4.5	2.04 1.93	−1.07 −0.98	1.06 1.04	0.87 0.92	2 1.9	1.06 1.01	0.87 0.91	2 1.4
TE	+40% +20%	4.6 4	4.9 4	2.08 2.11	−1.02 −1.11	1.11 1.08	0.89 0.86	2.5 2.4	1.16 1.09	0.94 0.86	2.7 2.4
	−20% −40%	4.4 5	4 4.7	2.14 2.11	−1.22 −1.30	1.06 1.03	0.82 0.78	2.4 2.6	1.06 1.02	0.82 0.72	2.4 2.8
KF	+40% +20%	4.8 4.5	4.6 4.1	2.29 2.21	−1.31 −1.24	1.1 1.08	0.73 0.81	3.5 3.2	1.11 1.08	0.77 0.81	3.4 2.9
	−20% −40%	4 4.3	4 4.4	2.04 1.97	−1.09 −1.02	1.06 1.02	0.88 0.91	2 1.5	1.06 1.03	0.88 0.94	2 1.7
TF	+40% +20%	4.5 4	4.7 4	2.02 2.08	−1.07 −1.13	107 1.07	0.92 0.87	1.7 2	1.07 1.07	0.91 0.87	1.8 2
	−20% −40%	4 4.4	4 4.1	2.18 2.27	−1.23 −1.28	1.07 1.07	0.80 0.72	2.4 2.8	1.07 1.07	0.81 0.74	2.4 2.9

presents the corresponding performance results under the same load disturbance conditions. It is noticed that the dynamic responses of ΔF1 and ΔF2 (in terms of both maximum overshoot and maximum undershoot), as well as the ΔPtie (in terms of settling time), exhibit minimal sensitivity to changes in AVR parameters. Additionally, the terminal voltage responses (ΔV1 and ΔV2) are only marginally affected by changes in AVR parameters.

In practical applications, control systems are often implemented in digital platforms, where feedback is inherently discrete due to sampling. Therefore, it is important to consider the implications of discrete or sampled-data feedback when analyzing stochastic control systems. As highlighted in [62], discrete-time formulations can exhibit different stability and performance characteristics compared to their continuous-time counterparts. While the current study adopts a continuous-time approach for theoretical clarity, the proposed methodology can be extended to discrete-time systems, enabling more realistic and implementable control strategies. The implicit nature of the proposed scheme, while enhancing numerical stability, introduces several computational challenges. It requires storing the full history of system states due to the nonlocal property of fractional derivatives, leading to increased memory consumption. Additionally, solving the system at each time step involves handling coupled nonlinear algebraic equations, which can be computationally intensive. These difficulties are managed using efficient sparse solvers and structured data handling, ensuring the trade-off between accuracy and computational cost remains acceptable for practical simulation purposes.

5. Conclusions

Coupling the AVR loop with the LFC via coupling coefficients facilitates simultaneous improvement of both frequency and voltage stability in an interconnected power system. In this study, loops of LFC and AVR were managed using a novel MSO-tuned (1+PDD²)-(1+TI) controller, used as a secondary regulator. Its dynamic performance was extensively evaluated on a conventional two-area power system. The proposed controller was optimized using various metaheuristic algorithms, including ChOA, DOA, PSO, GTO, GBO, and MSO, to enhance its performance. Among these, the MSO algorithm demonstrated superior results, delivering faster response times. The effectiveness of the (1+PDD²)-(1+TI) controller optimized via the MSO algorithm was benchmarked against other controllers—TID, FOPID, FOPI-PIDD², (1+PD)-PID, and the conventional PID—all of which were also tuned using the MSO algorithm.

Multiple scenarios were designed to assess the behavior of the (1+PDD²)-(1+TI) controller in managing frequency and tie-line power within a two-area hybrid power system. The controller was tested under diverse conditions, including sudden load changes at t = 0, random load fluctuations, common nonlinearities in power systems such as GRC and GDB, time-varying reference voltages in both areas, and ±20% and ±40% variations in system parameters. Across all scenarios, the (1+PDD²)-(1+TI) controller consistently outperformed its counterparts TID, FOPID, FOPI-PIDD², (1+PD)-PID, and the conventional PID controllers each optimized using the MSO algorithm. The proposed hybrid controller showed notable superiority in regulating frequency, tie-line power, and voltage deviations, demonstrating marked improvements in frequency stability and overall system performance compared to the other MSO-optimized controllers.

Our unique (1+PDD²)-(1+TI) controller, optimized using the MSO technique, demonstrated significant performance enhancements across key metrics. For frequency deviations (ΔF) and tie-line power (ΔPtie), it outperformed other MSO-based controllers, achieving improvements of 64% over FOPID, 62% over TID, and 60% over FOPI-PIDD₂. In terms of voltage deviation (ΔV) response, the proposed controller also showed superior performance, with improvements of 81% over TID, 76% over PID, 59% over FOPID, 20% over FOPI-PIDD², and 30% over the (1+PD)-PID controller. The results confirm the controller’s capability to mitigate disturbances and maintain system stability under dynamic scenarios. A sensitivity analysis further confirmed its robustness and adaptability across a broad range of operating scenarios.

Nevertheless, the controller poses certain challenges, such as higher computational demands compared to simpler alternatives, complexity in parameter tuning because of the high dimensionality, and the potential to underperform in comparison to “smart” controllers like fuzzy logic systems in certain cases. To overcome these drawbacks, several mitigation strategies can be employed. These include using faster processors to enhance computational efficiency, applying advanced tuning algorithms for optimal parameter selection, and exploring a hybrid optimization framework, where fuzzy logic will be employed to enhance the parameter tuning process of the (1+PDD²)-(1+TI) controller. Specifically, fuzzy inference systems will be used to dynamically adjust the search space or fine-tune initial parameter estimates, which will then be further optimized using metaheuristic algorithms like MSO. This integration aims to improve convergence speed, reduce computational burden, and enhance robustness in complex operating conditions. Moreover, the proposed controller has inherent limitations, particularly in terms of hardware complexity. It has not been evaluated under varying coupling and synchronization coefficients, and it neglects the impact of communication time delays.

Future research should concentrate on key aspects to improve applicability and validation of the proposed controller. These include addressing communication delays, extending the framework to multi-area power systems, integrating additional renewable energy sources, and employing real-time simulation platforms like OPAL-RT or dSPACE. Exploring these aspects will allow for a more comprehensive assessment of the controller’s performance and its potential deployment in complex and realistic power system environments.