What does this research mean for the field?

The proposed Hippopotamus Optimization-tuned sigmoid PID controller significantly outperforms existing PID-based metaheuristic controllers in load frequency control of a two-area thermal power system with renewable energy sources. Novelty: ClaimNovelty.NOVEL_FINDING. Consensus alignment: ConsensusAlignment.CHALLENGES_CONSENSUS.

What question did this study set out to answer?

To develop a sigmoid PID controller that adapts to variations in load frequency control in thermal power systems with renewable energy.

March 4, 2026Open Access

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

ÖCÖzay CanRecep Tayyip Erdoğan University MAMustafa Şinasi AyasKaradeniz Technical University AŞAli Kıvanç ŞahinErzurum Technical University

Key Points

To develop a sigmoid PID controller that adapts to variations in load frequency control in thermal power systems with renewable energy.
Proposed a sigmoid PID controller for load frequency control.
Utilized the Hippopotamus Optimization algorithm for tuning controller gains.
Evaluated performance under various scenarios including governor dead-band effects and renewable energy impacts.
Conducted frequency-domain stability analysis using Bode analysis.
The proposed controller significantly outperformed traditional PID controllers in minimizing ITAE and reducing settling time.
Demonstrated stable operation under parameter uncertainties and extreme variations in system parameters.
Achieved better control outcomes in terms of frequency stability compared to state-of-the-art methods.

Abstract

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.

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