Enhancing Reliability and Performance of Load Frequency Control in Aging Multi-Area Power Systems under Cyber-Attacks

Abstract

This paper addresses the practical issue of load frequency control (LFC) in multi-area power systems with degraded actuators and sensors under cyber-attacks. A time-varying approximation model is developed to capture the variability in component degradation paths across different operational scenarios, and an optimal controller is constructed to manage stochastic degradation across subareas simultaneously. To assess the reliability of the proposed scheme, both Monte Carlo simulation and particle swarm optimization techniques are utilized. The methodology distinguishes itself by four principal attributes: (i) a time-varying degradation model that broadens the application from single-area to multi-area systems; (ii) the integration of physical constraints within the degradation model, which enhances the realism and practicality compared to existing methods; (iii) the sensor suffers from fault data injection attacks; and (iv) an optimal controller that leverages particle swarm optimization to effectively balance reliability and system performance, thereby improving both stability and reliability. This method has demonstrated its effectiveness and advantages in mitigating load disturbances, achieving its objectives in just one-third of the time required by established benchmarks. The case study validates the applicability of the proposed approach and demonstrates its efficacy in mitigating load disturbance amidst stochastic degradation in actuators and sensors under FDIA cyber-attacks.

1. Introduction

The power system is becoming extremely challenging in terms of load demand changes and electricity quality, making loads highly unpredictable and uncertain, which has resulted in fluctuations in grid frequency and subsequently influenced the stability of the power system [1,2]. It is crucial to maintain the stability and reliability of interconnected power systems, preventing frequency deviations that can lead to power outages and damage to equipment [3,4,5]. For frequency and grid stabilization, Automatic Generation Control (AGC) is employed to adjust the output power of generators based on real-time conditions, ensuring that the power system can quickly return to a steady state after disturbances [6]. Load frequency control (LFC) [1,7] is proposed to regulate system frequency and manage power exchanges with neighboring areas. This mechanism is designed to maintain system frequency stays within acceptable boundaries and improve stability and reliability in power distribution, even as the power demand fluctuates [5]. Load disturbances are a constant presence, leading to deviations in the nominal system frequency and causing variations in the scheduled tie-line power flow between interconnected areas [3,4]. LFC systems based on Area Control Error (ACE) signals, which employ Proportional–Integral (PI) controllers [3,4], have been developed to enhance the dynamic response of power systems [5]. To address the uncertainties of the dynamic model and the nonlinearity of certain components (such as the governor and actuators, e.g., turbine), many related studies apply optimization algorithms to search for optimal controllers, such as meta-heuristic algorithms [8], ant algorithm [9], fuzzy logic [10,11], particle swarm optimization (PSO) [12] and genetic algorithm [13], neural networks [14], and deep learning [15].

The multi-area power systems, created by interconnecting several single-area systems, aim to improve the overall reliability and stability. They require the careful management of load and generation imbalances across different regions to maintain optimal performance [16,17,18,19,20]. Some control strategies, such as Active Disturbance Rejection Control, Sliding-Mode Control, and Model Predictive Control, have been applied to multi-area power systems, showcasing their advantages in handling disturbances introduced by renewable energy sources [21,22,23]. Furthermore, PSO has been used to tune PID controller parameters for LFC, demonstrating competitive performance in nonlinear scenarios [24]. Comparative studies of Genetic Algorithm, Simulated Annealing, and PSO for multi-area LFC systems further emphasize the potential of algorithms in enhancing control strategies [25]. However, a significant gap in the existing research is the lack of consideration of degraded components in the LFC of power systems. Degraded components, such as aging infrastructure and the wear and tear of mechanical parts, can significantly impact the performance and reliability of LFC systems [26]. Conventional control for LFC systems involves designing a fixed controller to manage load disturbances across various runtime conditions throughout the system’s lifetime [27,28,29]. However, since key components are aged, aging failures can degrade system performance [30]. Consequently, the degradation of key components can lead to diminished power generation precision and less accurate frequency measurements, impairing the reliability and stability of the LFC system [4,31].

Extensive experiments have shown that the component is incapable of delivering satisfactory performance once its degradation reaches a pre-defined threshold [32,33,34,35]. Many research studies describe actuator degradation using various models, such as a constant value, uniform distribution over predefined intervals, multi-state process, Gamma process, inverse Gaussian distribution, Wiener process, and dependent Markov chains [30,34,36,37,38]. However, these models primarily address single degradation processes, while real-world systems are often complex assemblies of different units, exhibiting multiple, potentially interrelated degradation processes due to varying operating and load conditions [39,40]. Specifically, many critical control system components, such as actuators, may experience varying degradation paths under different or even similar working conditions due to the inherent heterogeneity of components and systems [16,41]. It is essential to account for unit-to-unit variability, which results in different rates of degradation across components [42].

As the power grid expands, ensuring the security of all measurement and transmission components becomes increasingly difficult, and the cost of protecting grows prohibitively high [43]. In practical engineering applications, sensor degradation is not only driven by natural factors such as aging and intermittent failures but also by intentional cyber-attacks, including FDIA [44], Denial-of-Service (DoS) attacks [45], and deception attacks [46]. These threats compromise the accuracy and reliability of the data used for system control and decision-making. Consequently, sensor degradation models must incorporate both the gradual loss of sensor accuracy due to aging and the abrupt measurement errors introduced by cyber-attacks [16]. Monitoring sensor effectiveness is critical in preventing the entry of false or inaccurate data into control systems, which is essential for mitigating the risk of system-wide failures. Among various cyber-attacks, FDIA is regarded as the most significant and relevant because it specifically targets the integrity of state estimation by falsifying meter or sensor data, directly undermining control decisions that ensure system stability and security [47]. Ref. [48] proposed a mitigation strategy for DoS attacks in stable and well-regulated power systems by utilizing data prediction based on Deep Autoencoder Extreme Learning Machine (DAELM) models. A GAN-based mechanism was proposed, which detects and mitigates FDIA attacks by calculating, comparing, and adjusting control signals [49]. One study employed a Gaussian mixture model-based semi-supervised learning method to effectively detect and mitigate FDIA attacks on state estimation [50].

Physical limitations such as the Generation Rate Constraint (GRC) and the dead bands of speed governors critically affect the performance of LFC systems, which hinder the system’s ability to swiftly and accurately respond to frequency deviations, impacting its effectiveness in stabilizing load demand variations [5]. The thermal and mechanical properties inherent to the system components contribute to the nonlinear dynamics, thus adding complexity to accurately modeling the reliability functions in LFC systems across multiple areas when dealing with degraded elements. As a result, conventional reliability assessment techniques such as dynamic flow graph methodologies, Markov models, fault tree analysis, and dynamic Bayesian networks are inadequate for these scenarios [51,52,53]. Event-based Monte Carlo simulation (MCS) has proven to be an effective and data-driven method for assessing the reliability of intricate engineering systems, even in the absence of an exact system representation [54]. An analysis of the domain requirements is conducted to demonstrate how the system meets the necessary standards for real-time performance [55]. Without loss of generality, a multi-area load frequency control (LFC) system is assessed by its ability to meet the operational criteria. Therefore, the LFC reliability is defined by their capability to consistently achieve the required performance over a given period, even in the presence of various disturbances [56]. The estimated reliability of an LFC system can be enhanced by optimizing the parameters of the PI controller over a given period, a practice widely used in power systems [57,58]. Due to the nonconvex characteristics of the optimization problem, it is difficult to obtain the explicit expressions and value of global optimal parameters [59].

To overcome the above-mentioned challenges, this paper introduces a time-varying degradation model based on MCS and PSO to optimize LFC systems in multi-area settings and find the optimal solution for LFC controllers, such as PI and PID controllers. This approach copies the inherent complexity of real-world scenarios in which systems comprise multiple interconnected units. MCS is used to generate various possible operating environments and analyze the impact of different degradation levels on the system performance, thus evaluating the reliability of multi-area LFC systems with weakened components. PSO is utilized to design an optimal PI controller that strikes a balance between the reliability and the performance quality parameters, ensuring that the control strategies can adapt to component degradation and maintain system stability and reliability. This paper thoroughly analyzes key performance parameters, such as peak time, percentage overshoot, and settle time, to guarantee that the LFC system satisfies the operational requirements. This comprehensive assessment highlights the impact of degraded components on system performance and the effectiveness of the proposed control strategies. A case study illustrates that by incorporating stochastic degradation and cyber-attacks, and by extending the solution to multi-area power systems, the proposed methodology effectively addresses the coupling effects between Automatic Voltage Regulation and LFC loops across multiple areas.

The contributions can be summarized as follows:

• A time-varying degradation model considering FDIA is introduced for multi-area LFC systems, which extends from single-area to multi-area applications, effectively capturing the dynamic degradation characteristics and addressing the coupling effects between LFC loops and Automatic Voltage Regulation across different areas.
• The model incorporates physical constraints such as GRC and speed governor dead bands, providing a comprehensive representation of industrial conditions that are often neglected in the existing models.
• An optimal controller is designed using PSO, tailored to trade off reliability and performance quality metrics, which enhances both system stability and reliability.
• The proposed MCS-PSO method is validated through practical applicability to real-world multi-area LFC systems, demonstrating substantial improvements in reliability and performance over traditional control methods through simulation results.

This paper is organized as follows: Section 2 presents the model of multi-area LFC systems incorporating degradation and FDIA described by time-varying functions. Section 3 discusses the event-based MCS reliability evaluation and the performance enhancement of LFC systems through the application of the PSO method. Section 4 provides a case study of a three-area LFC system along with the corresponding results. Finally, the conclusions are drawn in Section 5, while future work is discussed in Section 6.

2. Dynamic Model of a Multi-Area Power System

2.1. Control Block Diagram of a Multi-Area LFC System

Frequency deviation within an area is a critical metric indicating the imbalance between power generation and load demand [5]. In multi-area power systems, regions are interconnected through tie lines, enabling the transfer of power between areas [60]. When imbalances occur, power exchange through these tie lines facilitates the restoration of frequency to its nominal level. Figure 1 outlines how this process is controlled within area i, where the Area Control Error ($AC{E}_i$) is expressed as a linear function of frequency deviations and power changes, ensuring that appropriate corrective measures are taken to maintain system stability as shown in the function below:

$$AC{E}_i=\Delta P_{ti{e}_i}+B_i\Delta f_i$$

(1)

2.2. Degradation Paths of Components under FDIA

The schematic of a control system with n degraded actuators and a degraded sensor suffering from FDIA is presented in Figure 2. The individual workload is not factored into the degradation model for actuators because it is heavily dependent on the total runtime $T_o$. The variations in workloads among different actuators and sensors within the same LFC system are minimal when compared to the total runtime $T_o$. For simplicity, this model considers that the degradation path of each component, such as actuators and sensors, is influenced solely by the total runtime $T_o$ and a few parameters to be estimated.

In the control system of Figure 2, which comprises n identical actuators, the degradation trajectory for each actuator, particularly the n-th one, can be described through a Wiener process to capture the differences in deterioration across the units [33]:

$$d{a}_n\left(T_0\right)=d{a}_0\left(T_0\right)+\lambda T_0+\sigma B\left(T_0\right)$$

(2)

where $d_{a_n}\left(T_0\right)$ is used to represent the loss in effectiveness of the n-th actuator after it has been operational for a duration $T_o$; $d{a}_n\left(0\right)$ stands for its initial degradation level where $d{a}_n\left(0\right)=0$; $\lambda$ stands for the stochastic parameter; and $\sigma B\left(T_o\right)\sim N(0,{\sigma }^2{T}_o)$ indicates the random dynamics in the degradation process.

Given the differing operational conditions of each actuator, it is crucial to acknowledge the variability in degradation paths from one unit to another. Consequently, it is logical to model $\lambda$ as following a normal distribution, represented as $\lambda \sim N({\mu }_{\lambda },{\sigma }_{\lambda }^2)$. Additionally, $\sigma$ is treated as a constant that characterizes the degradation process across similar types of components.

Assumption 1.

For any $\Delta t\ll T_0,d_{a_n}(T_o+\delta )=d_{a_n}\left(T_0\right)$. $\Delta t$ and $B\left(T_0\right)$ are independent.

In practical scenarios, the duration of the LFC process is considerably shorter than the total runtime of the control system. Consequently, the degradation of individual components is minimal during this brief task time. Therefore, the system output can be represented as follows:

$$Y\left(s\right)=U\left(s\right)Ga\left(s\right)(I-d_a\left(T_0\right))Gp\left(s\right)$$

(3)

where $Y\left(s\right)$ represents the Laplace transform of y, and $d_a\left(T_o\right)={[d_{a_1}\left(T_o\right),\cdots ,d_{a_n}\left(T_o\right)]}^{\mathrm{T}}$.

In industrial applications, sensor measurements are often influenced not just by sensor gain degradation $\left(d_s\left(T_0\right)\right)$ but also by measurement errors $\left(\omega \right)$ caused by potential cyber-attacks. The reduction in sensor performance over time, known as effectiveness loss, refers to the decline in accuracy at total runtime $T_0$ and can be represented as

$$ds\left(T_o\right)=ds\left(0\right)+\varsigma T_o+\sigma B\left(T_o\right)$$

(4)

where $d_s\left(0\right)$ represents the initial degradation state of the system, and $d_s\left(0\right)=0$ is defined as the initial degradation state, indicating that there is no degradation at the start of the process. Additionally, $\varsigma$ denotes the drift coefficient, which characterizes the rate of change in the degradation process over time. By defining the parameter $\varsigma$, we can model the dynamic behavior of degradation, facilitating accurate predictions of system longevity and maintenance schedule based on observed degradation trends. Therefore, Equation (4) reflects the unpredictable nature of the sensor gain degradation over time.

Assumption 2.

For any $\Delta t\ll T_0,d_s\left(T_o+\Delta t\right)=d_s\left(T_o\right)$. $d_s\left(T_0\right)$, ς and $d_a\left(T_0\right)$ are independent.

Thus, the sensor measurements transmitted to the controller under FDIA can be expressed as [50,61]

$$y_d=\left[1-d_s\left(T_0\right)\right]y+\omega$$

(5)

where $\omega$ is determined by the FDIA mechanism. Specifically, the attacker adapts the measurement tampering to resemble measurement errors, thereby avoiding detection. The attack vector $\omega$ is designed to follow a distribution similar to the measurement errors, denoted as $ϵ$. This formulation can be expressed as

$$\omega =arg\left[(\omega +ϵ)\sim \mathrm{distribution}\left(ϵ\right)\right]$$

(6)

It allows the sensor measurements to change while keeping the residue vector unchanged, making the attack undetectable. By disguising the attack as natural measurement noise, the attacker ensures that conventional anomaly detection systems, which rely on discrepancies in the residue, fail to identify the manipulation.

Since the frequency sensor is used in the supplementary loop [5], t is crucial to incorporate the sensor’s degradation dynamics into the supplementary control circuit. The degradation model for area i under FDIA is illustrated by Figure 3, where “Degra. (A)” stands for the degradation path of a turbine and “Degra. (S)” represents the degradation process of a sensor subject to FDIA.

3. Result Discussion and Assessment

3.1. Domain Requirement Analysis

The time domain requirements of LFC for multi-area systems are detailed in

Table 1. Time-domain requirements of LFC for multi-area systems.

Domain Requirements	Descriptions
Operational	Maximal peak time (PTmax)
	The time required for the result of the control process to reach its maximum or minimum value.
	Maximal percentage overshoot (POmax)
	The absolute value of the difference between the expected output and its maximum or minimum output, divided by the expected output.
	Maximal settling time (STmax)
	The duration from the initiation of the control process to the point when the output reaches and stays within a specific range.
Nonfunctional	Reliability
	The capacity of the LFC system to maintain its anticipated performance despite experiencing component degradation and cyber-attacks.

. It illustrates that multi-area LFC systems must satisfy several critical performance criteria. Key quality metrics include rising time (RT), declining time (DT), peak time (PT), percentage overshoot (PO), and settling time (ST), all of which are vital for evaluating real-time performance. Because the LFC process aims to return the frequency deviation to zero, RT and DT are not considered significant in this context.

In LFC systems, the quality parameters of PT, PO, and ST are crucial. PT ensures a rapid response to the command signal, PO minimizes oscillations during transient states, and ST guarantees that the system rapidly attains and sustains the desired output value, ensuring the stability of multi-area LFC systems. If any of these parameters exceed their respective maximum thresholds during the LFC process, the system is deemed to have failed in its control function.

In LFC systems, the key quality parameters PT, PO, and ST play a crucial role: (a) PT measures how quickly the LFC system responds to a command signal; (b) PO indicates how much the system’s response exceeds the desired value before settling, reflecting the system’s ability to minimize oscillations; and (c) ST measures how swiftly the system reaches and maintains the target output, ensuring stability. The LFC system is deemed to have failed if any of these parameters exceed their maximum allowable limits during operation.

In a single-area LFC system, failure is defined by the occurrence of any performance parameter exceeding its specified limits (PT, PO, or ST) or exceeding its corresponding maximal value during the control process:

$$\mathrm{P}\left(x=0\right)=\mathrm{P}(\mathrm{PF}>{\mathrm{PT}}_{max}\cup \mathrm{PF}>{\mathrm{PO}}_{max}\cup \mathrm{PF}>{\mathrm{ST}}_{max})$$

(7)

In a multi-area load frequency control (LFC) system comprising N control areas, the failure of the LFC is considered the occurrence of any performance parameter (PT, PO, or ST) exceeding its corresponding maximal value in any of the N control areas during the control process:

$$\mathrm{P}\left(x=0\right)=\mathrm{P}(x_1=0\cup x_2=0\cup \cdots x_N=0)$$

(8)

where

$$\mathrm{P}\left(x_1=0\right)=\mathrm{P}(\mathrm{PF}>{\mathrm{PT}}_{{max}_1}\cup \mathrm{PF}>{\mathrm{PO}}_{{max}_1}\cup \mathrm{PF}>{\mathrm{ST}}_{{max}_1})$$

(9)

$$\mathrm{P}\left(x_2=0\right)=\mathrm{P}(\mathrm{PF}>{\mathrm{PT}}_{{max}_2}\cup \mathrm{PF}>{\mathrm{PO}}_{{max}_2}\cup \mathrm{PF}>{\mathrm{ST}}_{{max}_2})$$

(10)

$\begin{array}{ccc}⋮& ⋮& ⋮\end{array}$

$$\mathrm{P}\left(x_N=0\right)=\mathrm{P}(\mathrm{PF}>{\mathrm{PT}}_{{max}_N}\cup \mathrm{PF}>{\mathrm{PO}}_{{max}_N}\cup \mathrm{PF}>{\mathrm{ST}}_{{max}_N})$$

(11)

Here, PF denotes the performance of the system output, and $x=0$ indicates a failed LFC process during simulation. When more stringent operational requirements are applied to the LFC system, its ability to maintain the desired system output can decline significantly. Consequently, selecting the appropriate operational requirements is crucial.

3.2. Event-Based MCS for Reliability Evaluation

The MCS method has been effectively employed for the reliability evaluation of stochastic systems, as it provides a straightforward and efficient approach [54]. The nonlinear dynamics associated with GRC and speed governor dead bands complicate the formulation of explicit reliability functions for multi-area LFC systems. Therefore, MCS must be used to model the operating environment of these systems, incorporating various degraded components to assess their reliability. The event-based MCS method is utilized to assess the reliability of control systems incorporating a PI controller [16]. The following Algorithm 1 is used to evaluate the reliability of the multi-area LFC system in accordance with the operational requirements.

Algorithm 1 Reliability estimation for multi-area LFC system.

Require: The PI controller’s parameters $K_{pi}$ and $K_{li}$ for area i ($i=1,2,\dots ,N$); the degradation model for a multi-area LFC system subject to FDIA; the operational requirements; the total runtime $T_0$. Ensure: The reliability estimation $R^{*}\left(T_0\right)$.  1: Set $m\leftarrow 0$.  2: Initialize H, l, $N_T$, $N_F$, $N_E$, $i\leftarrow 0$.  3: while $m<M$ do  4:    Perform a simulation using the dynamic model  5:    if simulation fails (Equation (7) or Equation (8)) then  6:      $N_F\leftarrow N_F+1$  7:      $R_i^{*}\left(T_0\right)=1-{\displaystyle \frac{N_F}{i}}$  8:    end if  9:    if $\left|R_i^{*}\left(T_0\right)-R_{i-1}^{*}\left(T_0\right)\right|/R_{i-1}^{*}\left(T_0\right)\le H$ then 10:      $N_E\leftarrow N_E+1$ 11:    end if 12:    if $N_E/i\le l$$i<N_T$ then 13:      $i\leftarrow i+1$ 14:    else 15:      $R^{*}\left(T_0\right)=1-N_E/N_T$ 16:      break 17:    end if 18:    $m\leftarrow m+1$ 19: end while 20: $R^{*}\left(T_0\right)={\sum }_{m=1}^M(R_m^{*}\left(T_0\right)/M)$

3.3. Reliability Enhancement via PSO Method

Kennedy and Eberhart [62] originally introduced the particle swarm optimization (PSO) technique, in which a solution is represented by a particle that encodes six parameters of the PI control strategy: $K_{P_1},K_{I_1},K_{P_2},K_{I_2},K_{P_3}$, and $K_{I_3}$.

Each particle’s fitness value in the optimization algorithm is calculated by assessing its performance through the application of a specifically designed fitness function. This function quantitatively evaluates how well a particle meets the criteria of the optimization problem, thereby determining its suitability within the search space. It is important to note that if the total runtime $T_0$ is still early in the system’s lifecycle, the reliability of the LFC system may be close to 1. This suggests that the component wear and tear is minimal at this stage [16].

In this situation, the PI controller must ensure that reliability remains at one while also optimizing the quality parameters (PT, PO, and ST) of the LFC process as much as possible. Consequently, two scenarios need to be considered:

• If the reliability is one, the fitness function of each particle will focus on optimizing the quality parameters of the LFC process.
• Conversely, the fitness function combines the reliability measure with a penalty for deviating from a reliability of one.

Based on the preceding discussions, the objective function for PSO is composed of two main components: the reliability-related component and the quality parameters-related component. These components are balanced using an equal penalty. Consequently, the fitness function for each particle is given by the following equation:

$$f\left(T_o\right)=\left(1-\Phi \left(R^{*}\left(T_o\right)\right)\right)·(1-R^{*}\left(T_o\right)+Q)+\Phi \left(R^{*}\left(T_o\right)\right)·W$$

(12)

where

$$\Phi \left(x\right)=\left\{\begin{array}{c}0,ifx\ne 1\\ 1,else\end{array}\right.$$

(13)

$$Q=a_1\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{PT}}_{{max}_i}+a_2\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{PO}}_{{max}_i}+a_3\sum _{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{ST}}_{{max}_i}$$

(14)

$$W=\sum _{m=1}^M\left({\displaystyle \frac{{\sum }_{i=1}^{N_T}\left(b_1·{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{PT}}_i+b_2·{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{PO}}_i+b_3·{\sum }_{\mathrm{i}=1}^{\mathrm{N}}{\mathrm{ST}}_i\right)}{N_T·M}}\right)$$

(15)

where $a_1$, $a_2$, and $a_3$ are weights to determine the value of the quality parameters-related part, and are set to 4, 10, and 0.3 based on a sufficient number of realistic simulations. Similarly, $b_1$, $b_2$, and $3_3$ are weights to determine the penalty function, and their values are set 2, 2, and 1.

Each particle in the swarm moves towards its best-known position, referred to as the local best, $p_{best}$, and towards the best position found by the entire swarm during the current iteration, known as the global best, $g_{bes}t$. For a particle i with j dimensions, its local best-known position is denoted as $p_{bes{t}_i}=(p_{i1},\dots ,p_{ij})$. The particle updates its velocity $v_{ij}^{k+1}$, and the position $x_{ij}^{k+1}$ using the following equations:

$$v_{ij}^{k+1}=w_1·t{v}_{ij}^k+c_1·{\rho }_1·\left(p_{ij}-x_{ij}^k\right)+c_2·{\rho }_2·\left(p_g-x_{ij}^k\right)$$

(16)

$$x_{ij}^{k+1}=x_{ij}^k+v_{ij}^k$$

(17)

where $k\in 1,\dots .,K;i\in 1,\dots .,I;j\in 1,\dots .,J$. $w_1$ indicates the inertia weight, $c_1$ and $c_2$ stand for the cognition learning factor and the social learning factor, respectively, and ${\rho }_1$ and ${\rho }_2$ are random factors limited to [0, 1].

The procedural steps of the proposed MCS-PSO methodology are illustrated in Figure 4.

4. Case Study

The case study system, illustrated in Figure 5, consists of three distinct control areas, each managed by different generation companies (Gencos). In each control area, three nonreheat turbine units are used for power generation. To simplify the modeling process, these units are represented as a single equivalent generator per control area, ensuring the essential dynamics of power generation and load balancing are maintained. The tie-line synchronizing coefficients, which regulate power exchange between the control areas in the three-area LFC system, are listed in

Table 2. Tie-line synchronizing coefficients.

Area	${\mathit{T}}_{\mathit{i}\mathit{j}}(\mathbf{p}\mathbf{.}\mathbf{u}\mathbf{.}/\mathbf{rad})$
1	$T_{12}=0.04,T_{13}=0.024$
2	$T_{21}=0.04,T_{23}=0.05$
3	$T_{31}=0.05,T_{32}=0.024$

. Additionally,

Table 3. Corresponding system parameters.

Parameters	Gencos
$MV{A}_{base}$	1	2	3	4	5	6	7	8	9
Rate (MW)	1100	900	1200	1000	800	1000	850	1000	1020
D (p.u./Hz)	0.014	0.014	0.014	0.015	0.015	0.015	0.016	0.016	0.016
$T_T$ (s)	0.4	0.36	0.42	0.3	0.4	0.41	0.3	0.4	0.41
$T_H$ (s)	0.08	0.06	0.07	0.08	0.06	0.07	0.08	0.06	0.07
R (p.u./Hz)	3	3	3	3	3	3	3	3	3
B (p.u./Hz)	1.014	1.014	1.014	1.015	1.015	1.015	1.016	1.016	1.016
$T_P$ (p.u. s)	0.15	0.15	0.15	0.166	0.166	0.166	0.1667	0.1667	0.1667
$\alpha$	0.6	0	0.4	0.4	0.4	0.2	0	0.5	0.5
Ramp rate (MW)	12	0	8	8	8	4	0	10	10

presents the system parameters, describing the operational characteristics of the generators and control mechanisms within the interconnected system.

Given the inherent nonlinearity and the degraded, time-varying characteristics of the 3-area LFC system under cyber-attack conditions, the model is implemented in MATLAB/SIMULINK instead of using a state-space representation. Traditional state-space representations, while useful for linear systems, often fail to capture the intricate dynamics and rapid changes induced by cyber threats, thus limiting their effectiveness in such applications. The system parameters are all expressed in per unit values, with a base of 1000 MW MVA. Each control area utilizes a PI controller. The system’s nominal frequency is 60 Hz, and the speed governor’s dead band is configured to a maximum value of 0.06%. In this scenario, significant step increases in demand are introduced at t = 0 s: $\delta P_{D_1}=-0.05\mathrm{p}.\mathrm{u}.(-50\mathrm{MW})$, $\delta P_{D_2}=0.1\mathrm{p}.\mathrm{u}.\left(100\mathrm{MW}\right)$, and $\delta P_{D_3}=0.06\mathrm{p}.\mathrm{u}.\left(60\mathrm{MW}\right)$. The regulation requirement for each area is established at 100 MW [60].

The degradation path of the control system, accounting for FDIA, is described as follows, with each time unit $T_0$ representing a period of two years. The degradation path for the three turbines $(n=3)$ is outlined as follows:

$$d_{a_n}\left(T_o\right)=d_{a_n}\left(0\right)+\lambda T_o+\sigma B\left(T_o\right)$$

(18)

where $\sigma B\left(T_o\right)\sim N\left(0,0.{005}^2·T_o\right)$ and $\lambda \sim N(0.01,0.002)$, which are general for turbines [63], specifically $d_{a_n}\left(0\right)=0$.

The degradation path of the frequency sensor subject to FDIA is

$$d_s\left(T_o\right)=d_s\left(0\right)+\varsigma T_o+\sigma B\left(T_o\right)$$

(19)

where $\varsigma =0.005$ and $\sigma B\left(T_o\right)\sim N\left(0,0.{002}^2·T_o\right)$, specifically $d_s\left(0\right)=0$; the FDIA implemented on the frequency sensor is $\omega \sim N(0,0.{01}^2·T_o)$. The degradation model of the above 3-area LFC system considering the effects of FDIA is therefore established.

4.1. Reliability Evaluation and Discussion

The 3-area LFC system is engineered to keep frequency deviations within ±0.3 Hz. To meet operational standards, the system must also satisfy the following quality criteria:

• $P{T}_{max}$: the output of the LFC process reaches its peak or trough within a maximum time of 1 s, 2 s, and 1 s.
• $P{O}_{max}$: In these three control areas, the maximum permissible frequency deviations are limited to 0.413%, 0.395%, and 0.365% of the nominal 60 Hz frequency.
• $S{T}_{max}$: the maximum time in these three required for the frequency to stabilize within 0.025% of the nominal 60 Hz frequency is 10.02 s, 8.67 s, and 10.35 s.

We have set the required number of Monte Carlo simulations, denoted as M, at 1000 to achieve a precision margin of 0.005 and a confidence level of 95%. The TA precision interval of 0.05 is used, ensuring that 95% of the simulations fall within this range. Each MCS run is performed in the MATLAB/SIMULINK environment, where each replication takes about 0.01 s. As a result, the MCS-PSO process takes approximately 7 h to compute the optimal PI controller settings for each total runtime $T_0$.

Table 4. The PI controller parameters of each area.

Area	${\mathit{K}}_{\mathit{P}}$	${\mathit{K}}_{\mathit{I}}$
1	$-6.96\times {10}^{-4}$	$-0.3435$
2	$-3.27\times {10}^{-4}$	$-0.3340$
3	$-1.60\times {10}^{-4}$	$-0.3398$

presents the parameters settings of the PI controllers, specifying their parameters for each of the three distinct control areas across various operational periods The quality parameters, assuming no component degradation, are evaluated based on the frequency–response curve shown in Figure 6.

Table 5. The quality parameters for each area without degradation or cyber-attacks.

Area	Peak Time	Percentage Overshoot	Settling Time
1	1.9 s	0.367%	8.59 s
2	0.6 s	0.402%	9.79 s
3	0.8 s	0.355%	8.02 s

provides the quality parameters for each control area under the conditions where there is no component degradation or FDIA cyber-attacks. The results demonstrate that the quality parameters of the three-area LFC system meet all operational requirements when degradation is absent. Following this, the reliability of the three-area LFC system is determined through event-based MCS. These simulations account for the total operational duration ($T_0$) and include the influences of both component degradation and FDIA cyber-attacks on system performance. Here, $T_0=0$ signifies that the system is free from degradation. Generally, the lifetime of the components of the power system can extend up to 25 years for most commercial products [30]. Consequently, in this case study, the time step for $T_0$ is set to 1 year, as degradation levels are unlikely to change significantly over a shorter period, such as a week, compared to the entire lifespan. Even with a one-week interval, the 7 h MCS-PSO process would still ensure the model’s applicability. The results are summarized in

Table 6. Reliability evaluation of the 3-area power system.

${\mathit{T}}_{\mathit{O}}$ Index	0	1	2	3	4	5	6	7	8	9	10	11
Reliability	1	1	1	1	1	1	1	0.9998	0.9961	0.9458	0.7336	0.3502
	1	1	1	1	1	1	1	0.9998	0.9961	0.9458	0.7336	0.3502

. A reliability value of 0.7336 indicates that there is a probability of 73. 36% that the LFC system will meet all operational requirements. It is noted that the reliability of the LFC system shows minimal variation with increasing total runtime $T_0$ during the first 6 years. This stability is due to the remaining performance of the system being sufficient to meet operational requirements with appropriate adjustments to the PI controller. However, reliability declines sharply as the system approaches the average lifespan of power systems, as the designed PI controller can no longer adequately compensate for the degradation effects through increased power output from the generators. This leads to a significant deterioration in the LFC process performance. Traditional control design methods struggle significantly with effectively managing component degradation.

4.2. Results for PI Controllers and Discussions

The MCS is applied to four different cases. These systems are built with identical internal control frameworks and similar component degradation. However, their differentiation lies in the control strategies implemented—parameters $K_{P_1},K_{I_1},K_{P_2},K_{I_2},K_{P_3}$, and $K_{I_3}$ within their PI controllers. Details of the PI controller parameters for each scenario are listed in

Table 7. Parameters of PI controllers used in each case.

Case	${\mathit{K}}_{\mathit{P}1}$	${\mathit{K}}_{\mathit{I}1}$	${\mathit{K}}_{\mathit{P}2}$	${\mathit{K}}_{\mathit{I}2}$	${\mathit{K}}_{\mathit{P}3}$	${\mathit{K}}_{\mathit{I}3}$
1	$-5.69\times {10}^{-4}$	$-0.3435$	$-8.86\times {10}^{-4}$	$-0.3340$	$-9.60\times {10}^{-4}$	$-0.3398$
2	$-6.96\times {10}^{-4}$	$-0.3435$	$-3.27\times {10}^{-4}$	$-0.3334$	$-1.60\times {10}^{-4}$	$-0.3398$
3	$-9.26\times {10}^{-3}$	$-0.3391$	$-7.36\times {10}^{-5}$	$-0.3298$	$-3.84\times {10}^{-5}$	$-0.3385$
4	$-1.38\times {10}^{-2}$	$-0.3672$	$-1.01\times {10}^{-4}$	$-0.3534$	$-5.96\times {10}^{-2}$	$-0.3746$

.

Event-based MCS simulations are next applied to each of the four cases, evaluating the effects of component degradation and FDIA cyber-attacks on system reliability across various total operational periods ($T_0$). The results are summarized in

Table 8. Reliability evaluation of each case in Table 7 as a function of $T_O$.

${\mathit{T}}_{\mathit{O}}$	Case
	0	1	2	3	4	5	6	7	8	9	10	11
1	1	1	1	1	1	1	1	1	0.9960	0.9460	0.7760	0.4062
2	1	1	1	1	1	1	1	0.9998	0.9961	0.9458	0.7336	0.3502
3	1	1	1	1	1	1	1	1	0.9963	0.9109	0.5433	0.2625
4	1	1	1	1	1	1	1	1	1	1	0.9980	0.9740

.

The reliability curves for each case are illustrated in Figure 7. Three key conclusions can be drawn from these results. Firstly, the event-based MCS method effectively estimates the reliability of the 3-area LFC system, utilizing the proposed time-varying degraded model. Secondly, systems with identical degraded components and structural configurations but different controller parameters exhibit varying reliability dynamics. Lastly, optimizing the controller parameters can significantly enhance the reliability of LFC systems in multi-area power systems.

4.3. MCS-PSO Optimization Results

The MCS-PSO method is applied to solve the reliability optimization problem for 3-area LFC systems. In this case study, each MCS run uses a sample size of 1000, and a total of 50 MCS replications are conducted.

In this study, 50 particles, each with 6 dimensions, are used, and 60 iterations are performed. Given that reliability values fall between 0 and 1, both the maximum velocity and maximum position are set to 1. The cognition learning factor and social learning factor are both set to 0.1 to align with the reliability range [0, 1]. The inertia weight is chosen as 0.6, based on the effectiveness of values between 0.9 and 0.4 in previous experiments.

The system reliability curves, comparing scenarios with and without optimization (refer to Table 6), are illustrated in Figure 7. The results indicate that optimized control strategies substantially enhance the reliability of LFC systems. Optimal control strategies achieve near-maximum reliability (close to 1) in all values of $T_0$. Compared to earlier designs, the proposed method demonstrates superior performance and increased reliability, effectively mitigating the impact of component degradation.

Table 9. The optimal parameters of PI controller.

Controller	${\mathit{T}}_{\mathit{O}}$
	0	1	2	3	4	5
$K_{P1},K_{I1}$	−0.1662,	−0.1603,	−0.1990,	−0.2000,	−0.1818,	−0.1742
	−0.2221	−0.2172	−0.2234	−0.2123	−0.2081	−0.2240
$K_{P2},K_{I2}$	−0.0729,	−0.0852,	−0.0601,	−0.0700,	−0.0771,	−0.0764
	−0.4805	−0.4940	−0.4840	−0.4891	−0.5255	−0.4665
$K_{P3},K_{I3}$	−0.1664,	−0.1520,	−0.1705,	−0.1880,	−0.1728,	−0.1492
	−0.1167	−0.1217	−0.1216	−0.1224	−0.1195	−0.1253
$f\left(T_O\right)$	27.6133	27.5518	27.5888	27.5918	27.6526	28.2008
$f\left(T_O\right)$ [60]	32.4716	32.5921	32.3587	32.4138	32.7917	33.0520
$K_{P1},K_{I1}$	−0.1722,	−0.1753,	−0.1663,	−0.1754,	−0.1730,	−0.1803
	−0.2210	−0.2092	−0.2296	−0.2163	−0.2333	−0.1947
$K_{P2},K_{I2}$	−0.0709,	−0.0683,	−0.0651,	−0.0493,	−0.0695,	−0.0608
	−0.4741	−0.4882	−0.4679	−0.4858	−0.4586	−0.5034
$K_{P3},K_{I3}$	−0.1568,	−0.1605,	−0.1504,	−0.1490,	−0.1637,	−0.1552
	−0.1265	−0.1144	−0.1256	−0.1200	−0.1236	−0.1136
$f\left(T_O\right)$	28.8556	28.9027	28.3671	28.5382	28.9161	28.9714
$f\left(T_O\right)$ [60]	36.4459	36.4962	33.1909	36.4422	36.7084	37.0918

lists the optimal parameters of each PI controller for total runtime $T_0\in 0,1,\dots ,11$ as determined by applying the MCS-PSO method to the time-varying model of the aging 3-area LFC system under cyber-attacks. Additionally, the table presents the fitness values for each optimal control strategy and the baseline strategy without optimization, across different $T_0$ values. The observed trend shows that as $T_0$ increases, the fitness value increases due to a decrease in reliability resulting from the degradation process. A smaller fitness value indicates better performance and reliability. The results indicate that the proposed controller outperforms the traditional controller in terms of efficacy. Specifically, in areas 2 and 3, the parameters $K_{P_1}$ and $K_{P_3}$ initially exhibit an increasing trend, which subsequently reverses into a decreasing trend, suggesting an adjustment to the system deviations. Conversely, in area 1, $K_{P_2}$ demonstrates an initial decreasing trend, followed by an increasing trend, reflecting a dynamic response to maintain control stability and effectiveness. This behavior is attributed to the interconnected nature of the three areas, which allows them to cooperate and compensate for the degradation process. $K_{I_1}$, $K_{I_1}$ and $K_{I_1}$ only exhibit minor variations across $T_0$, as they are less critical in determining the power output compared to $K_{P_1}$, $K_{P_2}$ and $K_{P_3}$ (proportional parameters). The simulation results further indicate that the MCS-PSO method significantly improves the reliability of multi-area LFC systems. This improvement is evident, as the fitness value is closely linked to the system’s reliability.

5. Conclusions

This paper presents an optimal control scheme designed to address the LFC problem in multi-area power systems affected by component degradation. A time-varying approximation model has been developed, capturing the complexity of degradation paths across various operational scenarios and offering a robust solution for managing stochastic degradation. Additionally, this study integrates considerations of FDIA, rather than other types of attacks, due to their common occurrence in LFC systems. The use of Monte Carlo simulation and particle swarm optimization provides a quantitative assessment of the system’s reliability. Integrating physical constraints within the model adds a layer of realism and enhances practical applicability, distinguishing this work from existing methods. The success demonstrated in case studies indicates significant potential for enhancing both the stability and reliability of multi-area power systems. Furthermore, this general framework is adaptable to a wide range of scenarios, and future research could explore the integration of additional cyber-attack types and corresponding mitigation techniques to further strengthen system resilience.

6. Future Work

Future research could explore several areas to further enhance the robustness and scalability of the proposed framework. A key direction for investigation is system generality, which may involve applying the framework to a broader range of power system configurations and addressing critical factors such as the influence of communication networks and delays. Additionally, advanced optimization techniques, including deep learning and evolutionary algorithms, could be explored to manage the complexities associated with large-scale power systems. The integration of acceleration techniques would also be essential to improve computational efficiency and scalability. Furthermore, the evaluation period of the degradation model could be extended by updating the drift parameter based on real-time health monitoring data from system components, ensuring that the model accurately reflects the realistic lifespan of these components.