## 4 Proposed Hybrid Model of NLMPC and Data-Driven Approach  
The control system presented in this research is modeled using a combination of physical state-space equations and data-driven methods. The data-driven approach can be employed to model complex, nonlinear relationships that are difficult to capture with first-principal models alone or to identify model parameters in real-time.  
In this study, a data-driven approach is incorporated into the NLMPC framework to enhance predictive accuracy and adaptability in AGC for multimachine power systems. The data-driven model is employed to approximate unmodeled system dynamics and disturbances, thereby improving the overall system response to uncertain operating conditions such as renewable energy fluctuations, cyber-physical attacks, and load variations.  
The data-driven component compensates for the inherent limitations of purely physics-based models, which may not fully capture nonlinear behaviors, system uncertainties, and transient disturbances. By learning from real-time operational data, the data-driven model estimates the discrepancy between predicted and actual system behavior and integrates this correction into the NLMPC framework.  
The proposed approach follows a structured implementation strategy to ensure accurate system predictions and effective control optimization. In the learning phase, real-time operational data, including generator states, control inputs, frequency deviations, and voltage fluctuations, are continuously collected. Machine learning techniques such as Artificial Neural Networks (ANNs), Gaussian Process Regression (GPR), and Long Short-Term Memory (LSTM) networks are employed to analyze system behavior and estimate unmodeled dynamics. The trained model is periodically updated using newly acquired data, allowing it to adapt to changing system conditions.  
In the prediction phase, the trained data-driven model predicts unmodeled system dynamics and refines the physical model's predictions. These refined predictions are integrated into the NLMPC optimization process, which balances generation cost, voltage stability, and frequency regulation while ensuring operational constraints are met. The control phase involves applying the optimized control inputs derived from the NLMPC framework to the system in real time. The data-driven model continuously updates and refines its predictions to ensure robust performance in dynamic environments.  
The integration of data-driven methods within the NLMPC framework provides several benefits. It improves accuracy by capturing complex system behaviors that traditional physics-based models may overlook. It enables adaptive learning by continuously updating using real-time data, ensuring responsiveness to evolving system conditions. Enhanced stability is achieved by improving the prediction of frequency and voltage deviations, thereby reducing control errors and system oscillations. Additionally, it optimizes generation costs while maintaining operational reliability and system security, ensuring economic efficiency.  
This hybrid approach effectively balances technical and economic objectives, making it highly suitable for real-time AGC in modern power systems. Future work may explore additional machine learning techniques to further refine the adaptability and efficiency of the proposed method.