# Bayat-driven FOPID controller design for biogas-based microgrid with real-time validation

## TL;DR

This paper proposes a Bayat-tuned FOPID controller for biogas-based microgrids to improve frequency stability under renewable energy fluctuations.

## Contribution

The novel contribution is the design and real-time validation of Bayat-driven FOPID controllers for decentralized microgrids with biogas and biodiesel sources.

## Key findings

- The Bayat-tuned FOPID controllers achieved a minimum rise time of 4.02 × 10⁻⁵ seconds and consistent settling time of ~49.8 seconds.
- FOPID-Bayat_SP2.0 outperformed other configurations with the lowest IAE, ITAE, ITSE, and ISE error indices.
- Real-time validation using OPAL-RT confirmed the practical effectiveness of the controllers in maintaining frequency stability.

## Abstract

The energy policies of the \documentclass[12pt]{minimal}
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\begin{document}$$21^{st}$$\end{document} century are increasingly focused on promoting generation solutions with minimal environmental impact. In response to strategic initiatives, the accelerating depletion of fossil fuel reserves has led to integrating renewable sources for power generation. The uncertain nature of solar and wind energy sources, along with fluctuating load demands, leads to frequency instability. This study addresses the challenge of frequency instability by designing a Bayat-tuned fractional-order proportional-integral-derivative (FOPID) controller for a decentralized microgrid \documentclass[12pt]{minimal}
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\begin{document}$$(Dz \mu G)$$\end{document}. The proposed \documentclass[12pt]{minimal}
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\begin{document}$$Dz \mu G$$\end{document} model consists of environmentally friendly energy sources such as a biogas turbine generator (BTG), a biodiesel engine generator (BEG), other distributed generation units (DGUs), and energy storage devices (ESDs). The mathematical modeling of \documentclass[12pt]{minimal}
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\begin{document}$$Dz \mu G$$\end{document} components is carried out using first-order transfer functions, which are combined to derive the overall transfer function of \documentclass[12pt]{minimal}
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\begin{document}$$Dz \mu G$$\end{document} model. This composite model is then approximated as a first-order plus time delay (FOPTD) system to simplify FOPID controller design. The parameters of the FOPID controller are optimized using the Bayat method to achieve robust performance under set-point tracking (SPT) and load disturbance rejection (LDR) scenarios. Based on this approach, three controller variants i.e., FOPID-\documentclass[12pt]{minimal}
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\begin{document}$$Bayat_{SP1.4}$$\end{document}, FOPID-\documentclass[12pt]{minimal}
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\begin{document}$$Bayat_{SP2.0}$$\end{document}, and FOPID-\documentclass[12pt]{minimal}
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\begin{document}$$Bayat_{LD1.4}$$\end{document}, are developed. To validate the effectiveness of the proposed control strategy, various simulation scenarios are considered, including load disturbances and varying levels of solar and wind power penetration. The performance of the controllers is evaluated in terms of frequency deviation, error mitigation, and transient behavior under SPT and LDR conditions. A comparative analysis using error indices, time-domain metrics, control effort, and frequency plots confirms the effectiveness of the Bayat-tuned FOPID designs. Furthermore, real-time validation using the OPAL-RT simulator underscores their practical potential in maintaining frequency stability within \documentclass[12pt]{minimal}
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\begin{document}$$Dz \mu G$$\end{document} systems. Owing to the performance analysis, it is justified that discussed FOPID–Bayat controllers consistently ensured controllability with a minimum rise time of \documentclass[12pt]{minimal}
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\begin{document}$$4.02 \times 10^{-5}\,\text {s}$$\end{document}, a nearly constant settling time of \documentclass[12pt]{minimal}
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\begin{document}$$\sim 49.8\,\text {s}$$\end{document}, and reduced control effort down to 0.12. Furthermore, error index evaluation confirmed that FOPID–Bayat\documentclass[12pt]{minimal}
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\begin{document}$$_{SP2.0}$$\end{document} outperformed other configurations by achieving the lowest IAE (8.737), ITAE (223.0), ITSE (40.39), and ISE (1.706), thereby demonstrating superior efficiency and robustness.