# Optimal Design of Fractional-Order PID Controllers for a Nonlinear AWS Wave Energy Converter Using Hybrid Jellyfish Search and Particle Swarm Optimization

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## Abstract

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## 1. Introduction

#### 1.1. Background and Motivation

#### 1.2. Literature Overview

#### 1.3. Objectives and Contributions

- A thorough mathematical modeling of the grid-connected AWS, including the back-to-back converter controllers, is presented, together with all of the system’s parameter values.
- The proposed FOPID controllers, the number of gains that must be tuned, and the HJSPSO method utilized for selecting the best gains are all detailed.
- The HJSPSO-FOPID controllers were compared with two conventional PID controllers that were tuned using PSO and COOT, in addition to FOPID controllers that were tuned using the GA.
- The controllers’ effectiveness and reliability were demonstrated by subjecting the grid-connected system to various unsymmetrical and symmetrical fault disturbances.

#### 1.4. Organization

## 2. Modeling of the AWS Wave Energy Conversion System

## 3. The Grid-Connected System: Block Diagram

#### 3.1. The Fractional PID (FOPID) Control Strategy

#### 3.2. The Back-to-Back Converter Configuration

## 4. Hybrid Jellyfish Search Optimizer and Particle Swarm Optimization (HJSPSO)

#### 4.1. HJSPSO Algorithm Steps

#### 4.2. HJSPSO Algorithm Computational Complexity

## 5. Nonlinear Grid-Connected AWS System Steady and Transient Responses

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Resultant elevation ($\eta \left(t\right)$) and (

**b**) excitation force of waves (${F}_{e}$).

**Figure 7.**(

**a**) $x$ and $v$ of the wave energy conversion device, (

**b**) the quadrature and direct axis currents of the stator, (

**c**) the three-phase generated currents (“${I}_{a}$” (black), “${I}_{b}$” (red), and “${I}_{c}$” (blue)), (

**d**) the real power produced by the linear generator, (

**e**) ${V}_{PCC}$ and ${V}_{DC}$, (

**f**) the injected real power (${P}_{PCC}$), and (

**g**) the injected reactive power (${Q}_{PCC}$).

**Figure 8.**(

**a**) ${V}_{PCC}$, (

**b**) ${V}_{DC}$, (

**c**) ${P}_{PCC}$, (

**d**) ${Q}_{PCC}$, and (

**e**) dq grid currents.

Parameter | Value | Parameter | Value | Parameter | Value |
---|---|---|---|---|---|

${m}_{f}$ | 4 × ${10}^{5}$ kg | γ | 1.4 | ${\psi}_{PM}$ | 23 Wb |

${m}_{add}$ | 3.55 × ${10}^{5}$ kg | θ | 4.5 m | $\lambda $ | 0.1 m |

ρ | 1025 kg/${\mathrm{m}}^{3}$ | ${H}_{s}$ | 4 m | ${C}_{DDW}$ | 0.4 |

${p}_{amp}$ | 1 × ${10}^{5}$ N/m^{2} | ${T}_{p}$ | 8 s | ${C}_{DUP}$ | 0.2 |

${\beta}_{wb}$ | 1.5 × ${10}^{6}$ kg/m | ψ | 4 m | R | 0.29 Ω |

$h$ | 43 m | μ | 0.1 | ${L}_{s}$ | 0.031 H |

${d}_{f}$ | 11 m | η | 0 | ${h}_{f}$ | 28.5 m |

${c}_{D}$ | 1 | ${c}_{M}$ | 2 | ${d}_{out}$ | 11 m |

$g$ | 9.8 m/${\mathrm{s}}^{2}$ | ${S}_{f}$ | 79 ${\mathrm{m}}^{2}$ | ${S}_{F}$ | 95 ${\mathrm{m}}^{2}$ |

Parameter | Value | Parameter | Value |
---|---|---|---|

DC link capacitance | 15 mF | Frequency | 50 Hz |

${Z}_{transformer}$ | j0.05 p.u | Base power | 1 MVA |

Filter’s resistance and inductance | R = 0.01 Ω and L = 0.0072 H | ${Z}_{pertransmissionline}$ | 0.02 + j0.14 p.u |

Controller | Algorithm | Gains |
---|---|---|

FOPID | HJSPSO | r = [519.27 438 0.065 1.022 0.95 542.46 792 0.24 0.82 0.93 4.22 61.31 0.14 0.32 0.27 0.32 3.44 1.33 0.004 0.19 19.92 2.01 0.14 1.05 0.34 0.11 0.22 0.88 0.18 0.33] |

FOPID | GA | r = [667.16 570.88 0.085 1.76 0.62 494.42 610.30 0.44 1.75 0.7 1.91 87.02 0.03 0.88 0.79 0.94 3.49 1.44 0.09 0.15 16.99 3.17 0.11 1.17 0.3 0.08 0.46 0.82 0.2 0.22] |

PID | COOT | r = [554 694.4 1 0 1 728 1355 1 0 1 7.6 90 1 0 1 1.17 21.8 1 0 1 2.88 239.1 1 0 1 2.43 27.8 1 0 1] |

PID | PSO | r = [374 878 1 0 1 747 1364 1 0 1 6.65 76 1 0 1 1.08 23.7 1 0 1 4.3 216.11 1 0 1 2.3 24.16 1 0 1] |

Point of Comparison (p.u) | PSO-PID | COOT-PID | GA-FOPID | HJSPSO-FOPID | Optimal Controller |
---|---|---|---|---|---|

Overshooting in ${V}_{PCC}$ | ~0.28 p.u | ~0.23 p.u | ~0.0 p.u | ~0.0 p.u | HJSPSO- and GA-FOPID |

Overshooting in ${V}_{DC}$ | ~0.03 p.u | ~0.03 p.u | ~0.007 p.u | ~0.006 p.u | HJSPSO-FOPID |

Undershooting in ${V}_{DC}$ | ~0.02 p.u | ~0.02 p.u | ~0.02 p.u | ~0.01 p.u | HJSPSO-FOPID |

Overshooting in ${Q}_{PCC}$ | ~0.23 p.u | ~0.5 p.u | ~0.3 p.u | ~0.1 p.u | HJSPSO-FOPID |

Undershooting in ${Q}_{PCC}$ | ~0.37 p.u | ~0.02 p.u | ~0.22 p.u | ~0.13 p.u | COOT-PID |

Point of Comparison (p.u) | PSO-PID | COOT-PID | GA-FOPID | HJSPSO-FOPID | Optimal Controller |
---|---|---|---|---|---|

Overshooting in ${V}_{PCC}$ during LG fault | ~0.16 p.u | ~0.25 p.u | ~0.2 p.u | ~0.0 p.u | HJSPSO-FOPID |

Undershooting in ${V}_{PCC}$ during LG fault | ~0.59 p.u | ~0.58 p.u | ~0.48 p.u | ~0.46 p.u | HJSPSO-FOPID |

Overshooting in ${V}_{PCC}$ during LLG fault | ~0.23 p.u | ~33% p.u | ~0.2% p.u | ~0.0 p.u | HJSPSO- and GA-FOPID |

Overshooting in ${V}_{PCC}$ during LL fault | ~0.3 p.u | ~0.36 p.u | ~0.05 p.u | ~0.0 p.u | HJSPSO-FOPID |

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**MDPI and ACS Style**

Ali, Z.M.; Ahmed, A.M.; Hasanien, H.M.; Aleem, S.H.E.A.
Optimal Design of Fractional-Order PID Controllers for a Nonlinear AWS Wave Energy Converter Using Hybrid Jellyfish Search and Particle Swarm Optimization. *Fractal Fract.* **2024**, *8*, 6.
https://doi.org/10.3390/fractalfract8010006

**AMA Style**

Ali ZM, Ahmed AM, Hasanien HM, Aleem SHEA.
Optimal Design of Fractional-Order PID Controllers for a Nonlinear AWS Wave Energy Converter Using Hybrid Jellyfish Search and Particle Swarm Optimization. *Fractal and Fractional*. 2024; 8(1):6.
https://doi.org/10.3390/fractalfract8010006

**Chicago/Turabian Style**

Ali, Ziad M., Ahmed Mahdy Ahmed, Hany M. Hasanien, and Shady H. E. Abdel Aleem.
2024. "Optimal Design of Fractional-Order PID Controllers for a Nonlinear AWS Wave Energy Converter Using Hybrid Jellyfish Search and Particle Swarm Optimization" *Fractal and Fractional* 8, no. 1: 6.
https://doi.org/10.3390/fractalfract8010006