## 4 Proposed Hybrid Model of NLMPC and Data-Driven Approach
### Implementation
Figures 5 and 7 highlight the proposed method's ability to maintain generator output voltage within acceptable limits:

* The hybrid approach consistently achieves voltage deviations below 0.2 pu across all tested scenarios, as seen in Table 2. For instance, during a line interruption between Bus 8 and 9, the voltage deviation remains at 0.137 pu, significantly lower than the 0.217 pu to 0.284 pu observed for other methods. * This low deviation ensures reliable power delivery and system resilience under disturbances. #### 5.1.4 Frequency Deviation

Frequency stability is vital for preventing cascading failures in power systems. Figure 6 demonstrates the proposed method's effectiveness in minimizing frequency deviations:

* During a fault scenario, the maximum frequency deviation achieved by the hybrid method is approximately 2.1% to 1.9%, as indicated in Table 2.
* Competing methods, however, show deviations ranging from 5.9% to 7.1% under similar conditions. This significant improvement underscores the method's ability to maintain system stability during dynamic disturbances. #### 5.1.5 Dynamic Response and Recovery

Figures 5-7 illustrate the hybrid method's superior dynamic response:* The proposed method achieves faster recovery from disturbances, with minimal oscillations in voltage and frequency. * Generator output power (Figure 7) stabilizes more quickly under the hybrid approach, ensuring efficient restoration of normal operating conditions. #### 5.1.6 Overall Performance Metrics

Table 2 consolidates the performance data for all metrics across various contingency scenarios. The proposed method consistently outperforms others, achieving the following:

* The lowest generation costs (by 5% to 10%);
* Minimal voltage deviations (below 0.2 pu);
* Reduced frequency deviations (approximately 2% to 3% lower);
* Superior turbine temperature regulation (close to 360 \({}^{\circ}\)C). These quantitative results highlight the hybrid NLMPC method's effectiveness in balancing technical and economic objectives, ensuring stability, reliability, and cost-efficiency in multimachine power systems under diverse operational challenges. ### IEEE Standard Model of 11 Bus ([PERSON]'s Two-Area System)

In this case study, the focus is on the standard model of the two-area, four-generator system, a well-established benchmark in power system stability studies [39]. This model is particularly significant for testing load frequency control (LFC) strategies, power system stabilizers (PSS), and other advanced control mechanisms. As illustrated in Figure 8, the schematic representation of this model, originally proposed by [PERSON] in his seminal book _Power System Stability and Control_, serves as a crucial tool for simulating the dynamic behavior of interconnected power systems under various disturbances. Building on the foundational principles outlined in earlier studies, this investigation seeks to enhance the performance of the proposed hybrid NLMPC and data-driven controller, specifically tailored for thermal turbines in synchronous generators. The primary objective is to achieve an efficient response that not only stabilizes the system but also ensures optimal and secure operation, even in the face of unexpected disruptions. By integrating the robust framework of the two-area, four-generator model with the innovative hybrid control approach, this study aims to push the boundaries of current methodologies. The enhanced controller is designed to rapidly adapt to real-time system conditions, providing superior stability and operational efficiency. This investigation focuses on evaluating the controller's performance under various disturbance scenarios, with particular attention paid to its effectiveness in minimizing system oscillations, regulating frequency, and maintaining thermal turbine efficiency. This approach is expected to yield significant improvements in power system stability and reliability, reinforcing the importance of advanced control strategies in modern power system management.