## 1 Introduction
These advancements collectively highlight how researchers are pushing the boundaries of what PID controllers can do, helping to smooth out frequency fluctuations, speed up response times, and keep power systems stable, even as renewable energy takes on a larger role. In addition to improving the PID controller as mentioned in the previous studies, optimizing the parameters of the PID controller through the integration of optimization algorithms is also a widely researched method. This method not only significantly improves LFC but also enhances the performance of the power system under various conditions. Optimization algorithms, such as Particle Swarm Optimization (PSO), Whale Optimization Algorithm (WOA), Gray Wolf Optimization (GWO), and many other techniques, have been applied to tune PID controllers, helping to achieve stability and improve frequency regulation. PSO has been widely used to optimize PID controllers in systems integrating thermal, hydro, and gas units. Studies have shown that PSO-based PID controllers outperform conventional methods, genetic algorithms (GAs), and differential evolution (DE) controllers by providing faster settling times and improved system responses. Notably, PID-PSO controllers have demonstrated performance improvements of up to 79% over the traditional techniques [15]. Additionally, PSO has been successfully employed in multi-source systems integrated with RES, addressing challenges such as load disturbances and ensuring better stability than other optimization methods [16; 17; 18]. Furthermore, PSO-optimizedPID controllers have been shown to outperform other advanced techniques like artificial neural networks (ANNs), offering minimal overshoot and achieving fast frequency recovery, making them a reliable choice for improving system stability in interconnected grids [19]. Moreover, PSO combined with the Gravitational Search Algorithm (GSA) has been applied to tune controllers for systems with nonlinearities, demonstrating remarkable improvements in stability and performance in both simulation and hardware-in-the-loop tests [20]. WOA, inspired by the humpback whale's bubble-net feeding strategy, has also proven effective in optimizing PID controllers for LFC. WOA has been applied to optimize fuzzy-PID and standard PID controllers in hybrid power systems, where it has shown superior frequency regulation even with parameter variations. Studies have demonstrated that WOA provides robust performance in systems with nonlinearities and disturbances, improving settling times and overall system stability [21; 22; 23]. Similarly, GWO has been used to optimize PID controllers in multi-source power systems, including those with renewable energy integration. Research has indicated that GWO significantly enhances frequency stability, reduces overshoot, and minimizes settling time when compared to traditional optimization methods, such as GA. GWO has shown its effectiveness in maintaining optimal frequency regulation in RES-integrated power systems, even amidst the fluctuations typical of renewable energy sources [24; 25]. Besides the algorithms above, other advanced optimization algorithms have also been applied to optimize PID controllers in power systems. For instance, the hybrid gravitational search and firefly algorithm (had) has been employed to optimize controllers in a two-area hydrothermal system, showing superior performance in handling load variations compared to conventional methods like PSO, GA, and GSA. This hybrid approach improved frequency regulation by offering better peak values and faster settling times [26]. Another technique is Artificial Bee Colony (ABC) optimization, this method has been used to enhance fuzzy logic-based PID control in isolated hybrid power systems, demonstrating significant improvements in power and frequency profiles, especially when integrated with superconducting magnetic energy storage (SMES) systems. The results showed better system stability and efficiency compared to other optimization techniques [27; 28].