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Optimal PID plus second-order derivative controller design for AVR system using a modified Runge Kutta optimizer and Bode’s ideal reference model

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Abstract

This paper presents the development of a new metaheuristic algorithm by modifying one of the recently proposed optimizers named Runge Kutta optimizer (RUN). The modified RUN (mRUN) algorithm is obtained by integrating a modified opposition-based learning (OBL) mechanism into RUN algorithm. A probability coefficient is employed to provide a good balance between exploration and exploitation stages of the mRUN algorithm. The greater ability of the mRUN algorithm over the original RUN algorithm is shown by performing statistical test and illustrating the convergence profiles. The developed algorithm is then proposed as an efficient tool to tune a proportional-integral-derivative (PID) plus second-order derivative (PIDD2) controller adopted in an automatic voltage regulator (AVR) system. The controlling scheme is further enhanced by integrating the Bode’s ideal reference model and using the performance index of integral of squared error as an objective function. The proposed reference model-based PIDD2 controller tuned by mRUN (mRUN-RM-PIDD2) approach is demonstrated to be superior in terms of transient and frequency responses compared to other available and best performing approaches reported in the last 5 years. In that respect, PID, fractional order PID (FOPID), PID acceleration (PIDA) and PIDD2 controllers tuned with the most effective algorithms reported in the last 5 years are adopted for comparisons. The comparative study confirms superior performance of the proposed method.

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Abbreviations

\(\% {\mathrm{OS}}\) :

Percent overshoot

ASO:

Atom search optimization

AVR:

Automatic voltage regulator

\(B_{W}\) :

Bandwidth

C-YSGA:

Chaotic yellow saddle goatfish algorithm

\(D\) :

Dimension size

\(D_{{\mathrm{M}}}\) :

Delay margin

\(E_{{{\mathrm{SS}}}}\) :

Steady state error

ECSA:

Enhanced crow search algorithm

EO:

Equilibrium optimizer

ESQ:

Enhanced solution quality

FOPID:

Fractional order proportional-integral-derivative

\(K_{{\mathrm{G}}}\) :

Generator gain

IWOA:

Improved whale optimization algorithm

HGSO:

Henry gas solubility optimization algorithm

\(k_{1}\), \(k_{2}\), \(k_{3}\) and \(k_{4}\) :

The coefficients of the search mechanism in Runge Kutta optimizer

\(K_{{\mathrm{A}}}\) :

Amplifier gain of the AVR system and acceleration gain of the PIDA controller

\(K_{{\mathrm{D}}}\) :

Derivative gain

\(K_{{{\mathrm{DD}}}}\) :

Second derivative gain

\(K_{{\mathrm{E}}}\) :

Exciter gain

\(K_{{\mathrm{I}}}\) :

Integral gain

\(K_{{\mathrm{P}}}\) :

Proportional gain

\(K_{{\mathrm{S}}}\) :

Sensor gain

\(L_{{\mathrm{l}}}\) :

Lower bound in Runge Kutta optimizer

LUS:

Local unimodal sampling

MRFO:

Manta ray foraging optimization

mRUN:

Modified Runge Kutta optimizer

\(N\) :

Number of solutions

OBL:

Opposition-based learning

\(P_{{\mathrm{G}}}\) :

Peak gain

\(P_{{\mathrm{M}}}\) :

Phase margin

\(P_{{{\mathrm{mOBL}}}}\) :

Probability coefficient in modified opposition-based learning

\(\varphi\) :

Random number

PID:

Proportional-integral-derivative

PIDA:

Proportional-integral-derivative and acceleration

PIDD2 :

Proportional-integral-derivative plus second-order derivative

\(Q_{{{\mathrm{indicator}}}}\) :

Quality indicator

\(r\) :

Random number

\({\mathrm{rand}}\) :

Random number

\({\mathrm{randn}}\) :

Random number with normal distribution

RM:

Reference model

RUN:

Runge Kutta optimizer

SA:

Simulated annealing

SF:

Adaptive factor in Runge Kutta optimizer

SFS:

Stochastic fractal search algorithm

SSA:

Salp swarm algorithm

\(T\) :

Total iterations

\(t\) :

Current iteration

\(t_{\max }\) :

Maximum iteration number

\(T_{{\mathrm{P}}}\) :

Peak time

\(T_{{\mathrm{R}}}\) :

Rise time

\(T_{{\mathrm{S}}}\) :

Settling time

TLBO:

Teaching learning-based optimization

\(U_{l}\) :

Upper bound in Runge Kutta optimizer

\(V_{{\mathrm{E}}}\) :

Error in voltage

\(V_{{{\mathrm{ref}}}}\) :

Reference input

\(V_{{\mathrm{S}}}\) :

Voltage measured by the sensor

\(V_{{\mathrm{T}}}\) :

Voltage of the synchronous generator

\(\omega_{{\mathrm{c}}}\) :

Crossover frequency

WOA:

Whale optimization algorithm

\(x_{{\mathrm{b}}}\) :

The best position represented in Runge Kutta optimizer

\(\overline{X}\) :

Opposite solution in opposition-based learning

\(x_{{{\mathrm{lbest}}}}\) :

The best solution in current iteration

\(x_{{\mathrm{w}}}\) :

The worst position represented in Runge Kutta optimizer

\(x_{r1} ,x_{r2}\) and \(x_{r3}\) :

Random solutions in Runge Kutta optimizer

\(\mu\) :

Random number

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No funding was received.

Author information

Authors and Affiliations

  1. Department of Electronics and Automation, Batman University, Batman, 72060, Turkey

    Davut Izci

  2. Department of Computer Engineering, Batman University, Batman, 72100, Turkey

    Serdar Ekinci

  3. Centre for Artificial Intelligence Research and Optimization, Torrens University, Adelaide, Australia

    Seyedali Mirjalili

  4. Yonsei Frontier Lab, Yonsei University, Seoul, South Korea

    Seyedali Mirjalili

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  1. Davut Izci
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Contributions

Serdar Ekinci and Davut Izci: Conceptualization, Methodology, Investigation, Writing – original draft, Visualization, Software. Seyedali Mirjalili: Writing - Review & Editing.

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Correspondence to Davut Izci.

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Izci, D., Ekinci, S. & Mirjalili, S. Optimal PID plus second-order derivative controller design for AVR system using a modified Runge Kutta optimizer and Bode’s ideal reference model. Int. J. Dynam. Control 11, 1247–1264 (2023). https://doi.org/10.1007/s40435-022-01046-9

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