## 1 Introduction  
Load frequency control (LFC) is a fundamental aspect of maintaining stability and reliability in power systems, ensuring that the balance between power generation and load demand is achieved while keeping system frequency within the prescribed limits [1]. Historically, LFC has relied on classical control strategies, such as proportional-integral-derivative (PID) controllers, which effectively manage the frequency regulation in conventional power systems. However, with the increasing complexity of modern power systems--driven by the integration of renewable energy sources (RESs) and the need for enhanced system flexibility--more advanced LFC approaches have been developed [2]. These include decentralized control strategies, such as those for turbine/governor systems [3], as well as emerging methods based on nonlinear models, like decentralized control using differential-algebraic equation (DAE) models [4] and sparse control techniques [5]. Such innovative approaches play a crucial role in enhancing system stability and reliability amidst the challenges of RES integration and fluctuating loads. The integration of RESs into power systems has greatly altered the dynamics of LFC. Unlike traditional power sources, which produce a consistent and predictable output, RESssuch as solar and wind energy are intermittent and variable, making it difficult to keep system frequency within the acceptable ranges. Several researchers have investigated this topic, emphasizing the importance of robust and adaptive LFC solutions to overcome the uncertainties and fluctuations inherent in renewable power supply. For example, ref. [6] developed a chaotic search-based hybrid optimization solution for LFC in a renewable energy-integrated power system, proving that the proposed method effectively reduced the influence of load changes and weather intermittency. Furthermore, ref. [7] examined several LFC methods for renewable energy-based hybrid power systems, highlighting the significance of modern and intelligent control approaches in ensuring system stability among the complicated interactions between conventional and renewable sources. In ref. [8], an adaptive LFC controller was presented that improved performance in the presence of disruptions induced by renewable energy sources such as wave energy conversion and photovoltaic systems. These studies highlight the growing relevance of creating LFC solutions that can accommodate the increasing penetration of RESs into power grids while providing stable and effective frequency control in modern energy systems. As power grids grow more complex, researchers have been working to make PID-based solutions for LFC more adaptable and reliable in modern systems. For example, one study [9] introduced a sliding mode-based state observer to handle uncertainties in multi-area, multi-source systems, giving PID controllers a performance boost. Another innovative approach [10] developed a hybrid fractional-order PID controller, fine-tuned using the manta ray foraging optimization algorithm, which proved to be effective at managing the frequency variations caused by renewable energy's unpredictable nature. Building on this, another paper [11] applied an Improved Ant Colony Optimization (IACO) technique to a fuzzy PID controller, enabling it to handle the nonlinearities of interconnected power systems. A different solution [12] used a hybrid simulated annealing-based optimizer to enhance a filtered PID controller, achieving excellent results in photovoltaic-thermal hybrid systems. Furthermore, a fractional-order fuzzy PID controller [13] tuned with a combination of adaptive differential evolution and pattern search excelled in navigating the challenges of multi-source interconnected systems. Finally, a cutting-edge framework [14] merged brain emotional learning with tilted integral derivative control, delivering impressive improvements in system stability and frequency regulation.