## 1 Introduction
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].