## 3 Result Discussion and Assessment  
### Domain Requirement Analysis  
The time domain requirements of LFC for multi-area systems are detailed in Table 1. 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.  
Figure 3: Dynamic model of the control area \(i\) with degraded components considering FDIA.  
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:  
\[\mathrm{P}(x=0)=\mathrm{P}(\mathrm{PF}>\mathrm{PT}_{\mathrm{max}}\cup\mathrm{PF}> \mathrm{PO}_{\mathrm{max}}\cup\mathrm{PF}>\mathrm{ST}_{\mathrm{max}}) \tag{7}\]  
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:  
\[\mathrm{P}(x=0)=\mathrm{P}(x_{1}=0\cup x_{2}=0\cup\cdots x_{N}=0) \tag{8}\]  
where  
\[\mathrm{P}(x_{1}=0)=\mathrm{P}(\mathrm{PF}>\mathrm{PT}_{\mathrm{max}_{1}}\cup \mathrm{PF}>\mathrm{PO}_{\mathrm{max}_{1}}\cup\mathrm{PF}>\mathrm{ST}_{\mathrm{ max}_{1}}) \tag{9}\]  
\[\mathrm{P}(x_{2}=0)=\mathrm{P}(\mathrm{PF}>\mathrm{PT}_{\mathrm{max}_{2}}\cup \mathrm{PF}>\mathrm{PO}_{\mathrm{max}_{2}}\cup\mathrm{PF}>\mathrm{ST}_{\mathrm{ max}_{2}}) \tag{10}\]  
\[\vdots\ \ \vdots\ \ \vdots\]  
\[\mathrm{P}(x_{N}=0)=\mathrm{P}(\mathrm{PF}>\mathrm{PT}_{\mathrm{max}_{N}}\cup \mathrm{PF}>\mathrm{PO}_{\mathrm{max}_{N}}\cup\mathrm{PF}>\mathrm{ST}_{\mathrm{ max}_{N}}) \tag{11}\]  
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.