## 4 Proposed Hybrid Model of NLMPC and Data-Driven Approach
### Implementation
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Disturbance**} & \multirow{2}{*}{**Indictors**} & \multirow{2}{*}{**Generation Cost**} & \multirow{2}{*}{**Maximum Frequency**} & \multirow{2}{*}{**Voltage**} & \multirow{2}{*}{**Temperature**} \\  & & **(M USD)** & & & **Deviation (pu)** & **(C\({}^{\circ}\))** \\ \hline \multirow{4}{*}{Interruption in line between Bus 6 and 9} & 409.652 & 3.3 & 0.112 & 362.982 \\  & PM & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & Ref Temp: 360 \\ \cline{2-6}  & \multirow{2}{*}{[31; 32]} & 445.694 & 5.9 & 1.12 & 533.37 \\  & & & \(\checkmark\) & \(\checkmark\) & Ref Temp: 360 \\ \cline{2-6}  & \multirow{2}{*}{[36; 37; 38]} & 435.609 & 6.1 & 0.196 & 451.202 \\  & & \(\boxtimes\) & \(\checkmark\) & \(\boxtimes\) & Ref Temp: 360 \\ \hline \multirow{4}{*}{Interruption in line between Bus 8 and 9} & 410.289 & 2.1 & 0.127 & 361.31 \\  & PM & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & Ref Temp: 360 \\ \cline{2-6}  & \multirow{2}{*}{[31; 32]} & 425.149 & 7.1 & 0.2781 & 551.28 \\  & & \(\boxtimes\) & \(\checkmark\) & \(\boxtimes\) & Ref Temp: 360 \\ \cline{2-6}  & \multirow{2}{*}{[36; 37; 38]} & 427.297 & 6.2 & 0.652 & 482.37 \\  & & \(\boxtimes\) & \(\checkmark\) & \(\boxtimes\) & Ref Temp: 360 \\ \hline \multirow{4}{*}{Interruption in line between Bus 8 and 9} & 419.279 & 1.9 & 0.1071 & 361.87 \\  & PM & \(\checkmark\) & \(\checkmark\) & \(\checkmark\) & Ref Temp: 360 \\ \cline{2-6}  & \multirow{2}{*}{[31; 32]} & 438.371 & 4.8 & 0.712 & 457.91 \\  & & \(\boxtimes\) & \(\checkmark\) & \(\boxtimes\) & Ref Temp: 360 \\ \cline{1-1} \cline{2-6}  & \multirow{2}{*}{[36; 37; 38]} & 435.627 & 5.1 & 0.8931 & 687.18 \\  & & \(\boxtimes\) & \(\boxtimes\) & \(\boxtimes\) & Ref Temp: 360 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Evaluation of proposed method (PM) and advanced techniques in various contingency scenarios on the IEEE 9-Bus system. Figure 7: Generator output power in the system with the proposed controller during the specified fault scenario. #### 5.1.1 Generation Costs

The total generation costs are a crucial metric for evaluating the economic efficiency of control methods. As shown in Figure 3, the proposed hybrid NLMPC approach achieves significantly lower generation costs compared to standard MPC and Adaptive MPC methods. For instance:

* **Pre-disturbance Costs**: The hybrid approach consistently maintains the lowest costs, highlighting its efficiency in normal operating conditions. * **Post-disturbance Costs**: Following a line outage or fault, the proposed method minimizes costs by rapidly adapting to new stable operating points, outperforming competitors. In Table 2, under various contingency scenarios (e.g., line interruptions between Bus 6 and 9 or Bus 8 and 9), the proposed method achieves generation costs of USD 410.29M and USD 419.28M, respectively, compared to USD 435.61M to USD 445.69M for alternative methods. This indicates a reduction in costs of approximately 5% to 8%, highlighting the method's economic advantage. #### 5.1.2 Turbine Temperature Regulation

Effective turbine temperature control is critical for avoiding thermal stress and ensuring system longevity. As shown in Figure 4, the hybrid method maintains turbine temperature at a closer level to the reference point (360 \({}^{\circ}\)C) than other methods do:

* Under a fault scenario, the hybrid approach keeps the temperature deviations minimal, with a post-disturbance value of 361 \({}^{\circ}\)C.
* By contrast, other methods exhibit larger deviations, with temperatures exceeding 450 \({}^{\circ}\)C or more, indicating poorer thermal regulation. This improved temperature stability reflects the proposed method's ability to integrate thermal constraints effectively into its optimization framework. #### 5.1.3 Voltage Deviation

Voltage regulation is a key indicator of system stability.