## 1 Introduction  
Maintaining grid stability in modern power systems is vital to ensure a reliable and high-quality electricity supply [1]. Power system stability refers to the ability of a power system to maintain operational equilibrium under normal conditions and to return to an acceptable equilibrium state after a disturbance [2]. Synchronous generators, which are pivotal in power plants, have to operate near their stability limits to meet fluctuating demand. Frequency stability is the balance between electrical demand and power generation [3, 4, 5]. A deviation from this balance can cause the system frequently to fall outside safe operating limits, threatening grid stability and security [6, 7, 8, 9]. Load frequency control (LFC) is essential for maintaining power system stability, particularly under fluctuation load conditions, by regulating the power balance between interconnected regions [10]. LFC ensures that the power generation matches the demand, preventing frequency deviation from its nominal value [11]. When this balance is disturbed, the frequency deviates, endangering both the security and stability of the power grid [12, 13, 14]. Automatic generation control (AGC) or LFC is responsible for correcting these imbalances and keeping the frequency and power within predefined limits. It is essential to maintain a reliable and high-quality electricity supply for consumers [15, 16, 17]. Power-frequency control operates at three distinct levels: primary, secondary, and tertiary. Each level has its own response time and associated control variables [18]. Primary frequency control responds within seconds (typically 2-30 s) and adjusts the generator's power output through droop control using speed governors [19]. This control mechanism helps stabilize the system by addressing the imbalance between generation and load. When the load increases, for example, the power output of the generator does not immediately adjust, and the required balance is achieved using the kinetic energy from the rotating generators. Speed controllers (governors) of each generator increase output to correct the speed drop and eliminate the imbalance [20]. Secondary control, operating between 30 s and 15 min, aims to bring the system back to its target value. It adjusts the frequency and power exchanges with neighboring regions using AGC [21]. Tertiary control works on a time scale of 10-30 min and involves adjusting unit commitments and the distribution of generation to restore primary and secondary reserves or optimize their distribution [22]. This level typically works on larger electrical systems, ensuring sufficient energy reserves and optimizing generation for economic reasons [23]. While traditional PI and PID controllers are widely used for their simplicity and efficiency, their performance can degrade in complex systems with fluctuating loads, leading to longer settling times and larger overshoots [24; 25; 26]. Various improvements to these controllers have been proposed to enhance their performance in LFC applications, including the use of hybrid algorithms like teaching-learning-based optimization (TLBO), imperialist competitive algorithm (ICA), and hybrid harmony search-cuckoo optimization to improve PID performance [27; 28]. Additionally, metaheuristic algorithms, such as the GBMO algorithm, fractional PID controllers, and Fuzzy PID controllers, have been developed to address the nonlinear and uncertain nature of power systems and improve controller robustness [29]. Other approaches include model predictive control (MPC) and event-triggered LFC strategies, which offer adaptive solutions to LFC in nonlinear, isolated, and interconnected microgrids [30]. PLCs are widely utilized in industrial automation systems, acting as controllers in a variety of applications, including motor control, water pumping, and switched reluctance motor control [31; 32; 33; 34]. Despite their simple design, PLCs can be combined with advanced control algorithms to handle complex control tasks efficiently [35].