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Optimising SMIB system stability: FOPID controller tuning via Harris hawks optimisation

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Abstract

Current power system stabilizer designs, including fuzzy-PID and Lead-Lag compensators, need help with adaptability and complexity in diverse and evolving power system environments. Conventional tuning methods like the Newton–Raphson approach and optimization strategies like particle swarm optimization, genetic algorithms, and cuckoo search algorithms face challenges in achieving optimal performance for Fractional Order Proportional Integral Derivative (FOPID) controllers. There is a critical need for innovative tuning methods that offer enhanced adaptability and performance in complex power system stability analysis. This research contributes to the advancement of power system stability analysis, specifically in SMIB systems, offering insights into optimizing FOPID controllers utilizing the innovative Harris Hawks Optimization (HHO) algorithm. The work expects these findings to broaden the implications for power system control and enhance the stability of Single Machine Infinite Bus (SMIB) systems, thus fostering the resilience and reliability of modern electrical infrastructure. The FOPID controller encompasses fractional order parameters, encompassing proportional, integral, and derivative gain, integral order, and derivative order, each exerting a substantial influence on control responses and stability. This research harnesses HHO, an optimization technique inspired by nature, to fine-tune FOPID parameters. The investigation involves initializing the SMIB model, formulating an objective function to minimize control errors, and employing HHO iteratively to refine the FOPID controller. The outcomes reveal enhanced stability, diminished overshoot, accelerated settling time, and transient response.

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Acknowledgements

The authors would like to express their gratitude to Department of Electrical and Electronics Engineering, Oriental University, Indore, Madhya Pradesh for all of their assistance and encouragement in carrying out this research and publishing this paper.

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Authors and Affiliations

Authors

Contributions

Yogesh Kalidas Kirange: Conceptualization, Methodology, Software, Validation, Formal Analysis, Investigation, Resources, Data Curation, Writing-Original Draft Preparation, Writing-Review Editing and Visualization. Pragya Nema: Supervision, Writing—original draft, Writing—review & editing.

Corresponding author

Correspondence to Yogesh Kalidas Kirange.

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Appendix: I

Appendix: I

Summary of abbreviations.

Symbol

Description

SI Unit

Vref

reference voltage signal for the generator’s excitation system

Volt

Vm

Nominal or rated voltage

Volt

KP

Proportional gain

Unitless

Δδ

Deviation in rotor angle from its equilibrium position

Radians

KI

Integral gain

Unitless

KD

Derivative gain

Unitless

t

Time

Seconds

Kp1

Fractional order proportional gain

Unitless

Ki1

Fractional order integral gain

Unitless

Kd1

Fractional order derivative gain

Unitless

α

Fractional order of the integral term

Unitless

β

Fractional order of the derivative term

Unitless

\({\text{P}}_{\text{e}}\)

Electrical power

Watt

\(\text{E}\) E

The internal voltage of the generator

Volt

\({\text{V}}_{\infty}\)

Voltage magnitude at the infinite bus

Volt

δ

Rotor angle of the generator

Radian

δ

Phase angle at the infinite bus

Degree

\({\text{X}}_{\text{d}}\)

Synchronous reactance

Ohm

Xd’

Transient reactance

Ohm

Xd’’

Sub transient reactance

Ohm

\(\text{M}\)

Rotor’s inertia

Kg-M2

\(\frac{dt}{d\omega }\)

Rate of change of angular velocity or angular acceleration

Rad/Sec2

\({\text{P}}_{\text{m}}\)

Mechanical power input

Watt

\({P}_{FOPID}\)

Power output from the FOPID controller

Watt

\({P}_{mech}\)

Mechanical power input

Watt

\(\text{D}\)

Damping coefficient

Unitless

Actual angular velocity

Rad/Sec

Ꞷs

Synchronous angular velocity

Rad/Sec

\(\frac{d\delta }{dt}\)

Rate of change of the rotor angle

Rad/Sec2

\(\text{e}\left(\text{t}\right)\)

Error signal at time \(\text{t}\) t

\(\Gamma\)

The gamma function computes a real-valued function’s integral over a given range

\({x}_{i}\left(t+1\right)\)

New position of the \({i}^{th}\) hawk

\({x}_{i}\left(t\right)\)

Current position of the \({i}^{th}\) hawk

\({V}_{i}\left(t+1\right)\)

Velocity or step size of the \({i}^{th}\) hawk at time t + 1

\(r\)

Random number

\({p}_{1}\), \({p}_{2}\)

Predefined probabilities

J(Kp1, Ki1, Kd1, α, β

The objective function to be minimised

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Kirange, Y.K., Nema, P. Optimising SMIB system stability: FOPID controller tuning via Harris hawks optimisation. Evol. Intel. 18, 16 (2025). https://doi.org/10.1007/s12065-024-00991-y

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