Hippopotamus optimization–tuned sigmoid PID controller for load frequency control of a two-area thermal power system with renewable energy sources

The increasing penetration of renewable energy sources (RES) in modern interconnected power systems introduces significant challenges for load frequency control (LFC) due to increased nonlinearities, the governor dead-band (GDB) effect, parameter uncertainties, and continuous power fluctuations. These factors reduce the effectiveness of conventional fixed-gain controllers and make it difficult to maintain frequency and tie-line power deviations within acceptable limits under varying operating conditions. To address these limitations, this study proposes a sigmoid proportional-integral-derivative (SPID) controller for LFC in a two-area thermal power system integrated with RES. In the proposed approach, the controller gains are adaptively adjusted through a sigmoid-based nonlinear mapping, and the corresponding parameters are optimally tuned using the Hippopotamus Optimization (HO) algorithm by minimizing the Integral of Time-weighted Absolute Error (ITAE) performance index. The performance of the proposed HO-SPID controller is evaluated under four time-domain scenarios: a baseline case without GDB and RES, a GDB-inclusive case without RES, a case with variable RES effects, and a robustness analysis of the baseline system under simultaneous ± 25% and ± 50% variations in all system parameters. Additionally, to demonstrate the stability of the proposed controller, frequency-domain stability is examined using Bode analysis. Simulation results demonstrate that the proposed HO-SPID controller significantly outperforms state-of-the-art PID-based metaheuristic controllers in terms of ITAE minimization, settling time, and undershoot, while the robustness analysis under parameter uncertainties and the Bode-based frequency-domain assessment jointly confirm its stable operation.