## 4 Proposed Hybrid Model of NLMPC and Data-Driven Approach
### Implementation
* **SOTA Methods**: Frequency deviations for other methods are significantly higher, often exceeding 5%, and in some cases reaching up to 7.1%. * **Analysis**: By maintaining smaller frequency deviations, the PM ensures improved system stability and faster recovery from disturbances. #### 5.2.4 Generation Costs

Economic efficiency is a key metric for evaluating the performance of control methods:

* **PM**: The PM consistently achieves the lowest generation costs, with values ranging from USD 1433.38 M to USD 1481.28 M across all scenarios. For example, during a line interruption between Bus 8 and 9, the PM incurs a cost of USD 1472.39 M, compared to USD 1577.32 M and USD 1623.71 M for SOTA methods. * **SOTA Methods**: Competing methods incur higher generation costs, often exceeding the PM's costs by 5% to 10%. * **Analysis**: The PM's ability to optimize generation costs demonstrates its superior economic performance, particularly under dynamic and adverse conditions. #### 5.2.5 Combined Evaluation

When assessing all metrics together, the proposed method consistently outperforms SOTA approaches. Key observations include the following:* **Turbine Temperature**: PM keeps deviations minimal, with temperatures consistently close to the reference value of 360 \({}^{\circ}\)C, whereas SOTA methods exhibit excessive deviations, posing risks to turbine longevity. * **Voltage and Frequency Deviations**: PM achieves the smallest deviations, ensuring stability and reliability during disturbances. * **Generation Costs**: PM minimizes operational costs, saving up to USD 150 M compared to alternative methods under similar scenarios. The proposed method excels across all key performance indicators:
* Maintains turbine temperatures within \(\pm\)2 \({}^{\circ}\)C of the reference value, significantly reducing thermal stress. * Ensures voltage deviations remain below 0.15 pu, minimizing instability risks. * Limits frequency deviations to below 4%, enhancing overall system resilience. * Reduces generation costs by 5-10% compared to state-of-the-art methods, ensuring economic efficiency. The quantitative results clearly demonstrate the superiority of the proposed hybrid NLMPC approach in balancing technical and economic objectives, making it the most effective and reliable method for multimachine power system control under challenging scenarios. The effectiveness of the proposed hybrid NLMPC approach with data-driven correction was evaluated through simulations on IEEE 9-bus and IEEE 11-bus test systems. The optimization process in the proposed method was formulated as a constrained nonlinear programming (NLP) problem and solved using Sequential Quadratic Programming (SQP). This choice was made due to its ability to handle complex nonlinear constraints while ensuring computational efficiency in real-time AGC applications. The average computation time required to obtain the control solution per iteration ranged between 85 and 120 milliseconds, making it suitable for real-time operation. Compared to existing MPC-based AGC methods reported in references [24; 25] and [29; 30; 31], which require 180-250 milliseconds per iteration, the proposed approach significantly reduces computational burden by 30-50%. This improvement is primarily attributed to the integration of the data-driven correction mechanism, which enhances predictive accuracy while reducing optimization complexity. The proposed method achieves faster convergence by approximating unmodeled system dynamics, thereby enabling efficient and adaptive control actions. Furthermore, the simulation results demonstrate that the proposed method not only reduces computational time but also improves control performance by minimizing frequency deviations, voltage fluctuations, and generation cost more effectively than conventional methods. The NLMPC framework, integrated with data-driven learning, dynamically adapts to system variations and external disturbances, ensuring a robust and stable power system operation.