## 2 Dynamic Model of a Multi-Area Power System
### 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]:  
\[da_{n}(T_{0})=da_{0}(T_{0})+\lambda T_{0}+\sigma B(T_{0}) \tag{2}\]  
Figure 1: Control block diagram of area \(i\). where \(d_{a_{n}}(T_{0})\) is used to represent the loss in effectiveness of the \(n\)-th actuator after it has been operational for a duration \(T_{o}\); \(da_{n}(0)\) stands for its initial degradation level where \(da_{n}(0)=0\); \(\lambda\) stands for the stochastic parameter; and \(\sigma B(T_{o})\sim N(0,\sigma^{2}T_{o})\) 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. For any \(\Delta t\ll T_{0}\), \(d_{a_{n}}(T_{o}+\delta)=d_{a_{n}}(T_{0})\). \(\Delta t\) and \(B(T_{0})\) 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:  
\[Y(s)=U(s)Ga(s)(I-d_{a}(T_{0}))Gp(s) \tag{3}\]  
where \(Y(s)\) represents the Laplace transform of \(y\), and \(d_{a}(T_{o})=[d_{a_{1}}(T_{o}),\cdots,d_{a_{n}}(T_{o})]^{\mathrm{T}}\). In industrial applications, sensor measurements are often influenced not just by sensor gain degradation \((d_{s}(T_{0}))\) but also by measurement errors \((\omega)\) caused by potential cyberattacks. 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  
\[ds(T_{o})=ds(0)+\varsigma T_{o}+\sigma B(T_{o}) \tag{4}\]  
where \(d_{s}(0)\) represents the initial degradation state of the system, and \(d_{s}(0)=0\) 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. For any \(\Delta t\ll T_{0}\), \(d_{s}(T_{o}+\Delta t)=d_{s}(T_{o})\). \(d_{s}(T_{0})\), \(\varsigma\) and \(d_{a}(T_{0})\) are independent. Figure 2: Structure of the control system subject to degradation and FDIA. Thus, the sensor measurements transmitted to the controller under FDIA can be expressed as [50; 61]  
\[y_{d}=[1-d_{s}(T_{0})]y+\omega \tag{5}\]  
where \(\omega\) is determined by the FDIA mechanism.