# A New MPPT-Based Extended Grey Wolf Optimizer for Stand-Alone PV System: A Performance Evaluation versus Four Smart MPPT Techniques in Diverse Scenarios

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## Abstract

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## 1. Introduction

#### 1.1. Motivations

#### 1.2. State of the Art

#### 1.3. Contributions

#### 1.4. Structure Overview

## 2. PV System Plant Modeling

#### 2.1. PV Mathematical Model

**Sun Earth Solar Power TDB156x156-60-P 215W**) under standard test conditions (STC) considered in the simulation are listed in Table 1.

#### 2.2. DC/DC Boost Converter

## 3. MPPT Control Method

#### 3.1. Extended and Grey Wolf Optimizers

#### 3.1.1. GWO Mathematical Model

- Social Hierarchy: The social hierarchy of the wolves is represented by four positions: ${X}_{\alpha}$, ${X}_{\beta}$, ${X}_{\delta}$ and ${X}_{\omega}$. These positions represent the best, second-best, third-best and the rest of the wolves in the population, respectively.
- Encircling behavior: Entails the coordinated movement of group members toward a specific target position while tightening the search area around it. The wolves concentrate their exploration around influential leaders ($\alpha $, $\beta $, $\delta $), optimizing the balance between exploration and exploitation. This approach enables the algorithm to effectively discover optimal or near-optimal solutions for complex optimization problems. The encircling behavior equations are given as follows [49]:$$\overrightarrow{D}=|\overrightarrow{C}\xb7\overrightarrow{{X}_{p}}\left(t\right)-\overrightarrow{X}\left(t\right)|$$$$\overrightarrow{X}(t+1)=|\overrightarrow{{X}_{p}}\left(t\right)-\overrightarrow{A}\xb7\overrightarrow{D}|$$$$\overrightarrow{A}=2\overrightarrow{a}\xb7{\overrightarrow{r}}_{1}-\overrightarrow{a}$$$$\overrightarrow{C}=2\xb7{\overrightarrow{r}}_{2}$$
- Follow, hunt and approach the prey: The $\alpha $, $\beta $, $\delta $ wolves guide the $\omega $ wolves toward promising regions. The updated position for each omega wolf is determined by the influence of the $\alpha $, $\beta $ and $\delta $ wolves as follows:$$\overrightarrow{{D}_{\alpha}}=|\overrightarrow{{C}_{1}}\overrightarrow{{X}_{\alpha}}\left(t\right)-\overrightarrow{X}\left(t\right)|,{\overrightarrow{X}}_{1}=\overrightarrow{{X}_{\alpha}}-\overrightarrow{{A}_{1}}\xb7\overrightarrow{{D}_{\alpha}}$$$$\overrightarrow{{D}_{\beta}}=|\overrightarrow{{C}_{2}}\overrightarrow{{X}_{\beta}}\left(t\right)-\overrightarrow{X}\left(t\right)|,{\overrightarrow{X}}_{2}=\overrightarrow{{X}_{\beta}}-\overrightarrow{{A}_{2}}\xb7\overrightarrow{{D}_{\beta}}$$$$\overrightarrow{{D}_{\delta}}=|\overrightarrow{{C}_{3}}\overrightarrow{{X}_{\delta}}\left(t\right)-\overrightarrow{X}\left(t\right)|,{\overrightarrow{X}}_{3}=\overrightarrow{{X}_{\delta}}-\overrightarrow{{A}_{3}}\xb7\overrightarrow{{D}_{\delta}}$$Therefore, the updated position of all search agents is given by [49]:$$\overrightarrow{X}(t+1)=\frac{\overrightarrow{{X}_{1}}+\overrightarrow{{X}_{2}}+\overrightarrow{{X}_{3}}}{3}$$

#### 3.1.2. EGWO Mathematical Model

#### 3.1.3. EGWO and GWO Application for MPPT

**Initialization**: The optimization process starts with the initialization of a population ${N}_{p}$ (represented by wolves) in the search space. The duty ratio ${d}_{i}$ is initialized (Equation (18)) randomly within the defined limits, ranging from $0.1$ to $0.9$.$${d}_{i}=rand({N}_{p},1)({d}_{max}-{d}_{min})+{d}_{min}$$**Evaluation**: The fitness values, corresponding to the PV power output, are calculated for each member of the population. The wolves with the highest PV power values are assigned as $d\alpha $ (the best solution), $d\beta $ (the second-best solution) and $d\delta $ (the third-best solution).**Updating Positions**: The positions ${d}_{i}$ (duty ratios) of the wolves in the population are updated based on the positions of $d\alpha $, $d\beta $ and $d\delta $, the best, second and the third-best solutions, respectively. This update aims to explore the search space more effectively and improve the duty ratios. The updated position of all search agents using GWO and EGWO is given as follows [52]:$$\overrightarrow{{D}_{\alpha}}=|\overrightarrow{{C}_{1}}\overrightarrow{{d}_{\alpha}}-\overrightarrow{{d}_{i}}|,{\overrightarrow{d}}_{1}=\overrightarrow{{d}_{\alpha}}-\overrightarrow{{A}_{1}}\xb7\overrightarrow{{D}_{\alpha}}$$$$\overrightarrow{{D}_{\beta}}=|\overrightarrow{{C}_{2}}\overrightarrow{{d}_{\beta}}-\overrightarrow{{d}_{i}}|,{\overrightarrow{d}}_{2}=\overrightarrow{{d}_{\beta}}-\overrightarrow{{A}_{2}}\xb7\overrightarrow{{D}_{\beta}}$$$$\overrightarrow{{D}_{\delta}}=|\overrightarrow{{C}_{3}}\overrightarrow{{d}_{\delta}}-\overrightarrow{{d}_{i}}|,{\overrightarrow{d}}_{3}=\overrightarrow{{d}_{\delta}}-\overrightarrow{{A}_{3}}\xb7\overrightarrow{{D}_{\delta}}$$$$\overrightarrow{{d}_{i}}(t+1)=\frac{\overrightarrow{{d}_{1}}+\overrightarrow{{d}_{2}}+\overrightarrow{{d}_{3}}}{3}$$$$\overrightarrow{{d}_{i}}(t+1)=\frac{{\alpha}_{em}\overrightarrow{{d}_{1}}+{\beta}_{em}\overrightarrow{{d}_{2}}+{\delta}_{em}\overrightarrow{{d}_{3}}}{3}$$**Termination**: The termination condition is determined by the maximum number of iterations reached or when the relative change in PV power compared to the previous iteration’s power becomes negligible. The termination criterion is defined as follows:$$\frac{|{P}_{pv}-{P}_{pv,old}|}{{P}_{pv,old}}\ge \Delta p$$

## 4. Simulation Results

**Sun Earth Solar Power TDB156x156-60-P 215W**) under different irradiance levels and temperatures. Additionally, the proposed MPPT parameters are listed in Table 3.

#### 4.1. First Scenario: Under Standard Test Conditions (STC)

#### 4.2. Second Scenario: Variable Irradiance and Constant Temperature

#### 4.3. Third Scenario: Variable Temperature and Constant Irradiance

#### 4.4. Fourth Scenario: Variable Temperature and Irradiance

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPPT | Maximum power point tracking |

MPP | Maximum power point |

EGWO | Extended grey wolf optimizer |

GWO | Grey wolf optimizer |

EOA | Equilibrium optimization algorithm |

PSO | Particle swarm optimization |

SCA | Sin cos algorithm |

P&O | Perturb and observe |

INC | Incremental conductance |

FOCV | Fractional open circuit voltage |

FSCC | Fractional short circuit current |

HL | Hill climbing |

ABC | Artificial bee colony algorithm |

GA | Genetic algorithm |

ACO | Colony optimization |

FA | Firefly algorithm |

WOA | Whale optimization algorithm |

CS | Cuckoo search |

AFSA | Artificial fish swarm algorithm |

PWM | Pulse width modulation |

d | Duty cycle |

P | PV system power (W) |

${I}_{out}$ | Cell current |

${I}_{ph}$ | Current generated by light |

${I}_{d}$ | Diode’s current |

${I}_{sh}$ | Current of the parallel resistance |

${I}_{0}$ | Reverse saturation current |

V | Voltage across the PV cell |

${V}_{d}$ | Voltage of the equivalent diode |

${R}_{s}$ | Series resistance |

${R}_{sh}$ | Parallel resistance |

G | Actual irradiance |

${G}_{STC}$ | Irradiance at standard rating conditions |

K | Boltzmann constant |

q | Electron charge |

${T}_{c}$ | Cell temperature |

${T}_{r}$ | Reference temperature |

n | Diode ideality factor |

${N}_{s}$ | Number of series cells |

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**Figure 2.**Extended grey wolf optimizer (EGWO) maximum power point tracking (MPPT) flowchart [52].

Specifications | Value |
---|---|

Maximum power (W) | $215.028$ |

Cells per module (${N}_{\mathit{cell}}$) | 60 |

Open circuit voltage ${V}_{\mathit{oc}}$ (V) | $36.8$ |

Short-circuit current ${I}_{\mathit{sc}}$ (A) | $7.92$ |

Voltage at maximum power point ${V}_{\mathit{mp}}$ (V) | $29.7$ |

Current at maximum power point ${I}_{\mathit{mp}}$ (A) | $7.24$ |

Temperature coefficient of ${V}_{\mathit{oc}}$ (%/deg.C) | $-0.34$ |

Temperature coefficient of ${I}_{\mathit{sc}}$ (%/deg.C) | $0.05$ |

Specifications | Value |
---|---|

Inductance (L) | $0.15$ H |

In capacitor (C) | 100 × 10${}^{-6}$ F |

Out capacitor (C) | 470 × 10${}^{-6}$ F |

Max ${f}_{SW}$ | 10 kHz |

Load | 32 Ohm |

Parameter | EGWO | GWO | PSO | EOA | SCA |
---|---|---|---|---|---|

Population size (${N}_{p}$) | 20 | 20 | 20 | 20 | 20 |

Maximum number of iterations (${t}_{max}$) | 100 | 100 | 100 | 100 | 100 |

A and C | Random | Random | − | − | − |

${\alpha}_{em}$ | $1.5$ | − | − | − | − |

${\beta}_{em}$ | $1.2$ | − | − | − | − |

${\delta}_{em}$ | $1.1$ | − | − | − | − |

${w}_{inertia}$ | − | − | $0.7$ | − | − |

${C}_{cognitive}$ | − | − | $1.4$ | − | − |

${C}_{social}$ | − | − | $1.4$ | − | − |

${a}_{1}$ | − | − | − | 2 | − |

${a}_{2}$ | − | − | − | 1 | − |

$GP$ | − | − | − | $0.5$ | − |

${r}_{1,2,3,4}$ | − | − | − | − | Random in $[0,1]$ |

Algorithm | Scenario 1 (Oscillation) | Scenario 2 (Oscillation) | Scenario 3 (Oscillation) | Scenario 4 (Oscillation) |
---|---|---|---|---|

EGWO | Excellent (0.09 W) | Excellent (0.09 W) | Excellent (0.09 W) | Excellent (0.09 W) |

GWO | Good (0.266 W) | Good (0.266 W) | Good (0.266 W) | Good (0.266 W) |

PSO | Poor (0.731 W) | Poor (18.401 W) | Poor (1.919 W) | Poor (37.901 W) |

EOA | Poor (1.044 W) | Poor (26.671 W) | Poor (2.035 W) | Poor (39.323 W) |

SCA | Poor (0.729 W) | Poor (15.796 W) | Poor (2.035 W) | Poor (35.379 W) |

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## Share and Cite

**MDPI and ACS Style**

Silaa, M.Y.; Barambones, O.; Bencherif, A.; Rahmani, A.
A New MPPT-Based Extended Grey Wolf Optimizer for Stand-Alone PV System: A Performance Evaluation versus Four Smart MPPT Techniques in Diverse Scenarios. *Inventions* **2023**, *8*, 142.
https://doi.org/10.3390/inventions8060142

**AMA Style**

Silaa MY, Barambones O, Bencherif A, Rahmani A.
A New MPPT-Based Extended Grey Wolf Optimizer for Stand-Alone PV System: A Performance Evaluation versus Four Smart MPPT Techniques in Diverse Scenarios. *Inventions*. 2023; 8(6):142.
https://doi.org/10.3390/inventions8060142

**Chicago/Turabian Style**

Silaa, Mohammed Yousri, Oscar Barambones, Aissa Bencherif, and Abdellah Rahmani.
2023. "A New MPPT-Based Extended Grey Wolf Optimizer for Stand-Alone PV System: A Performance Evaluation versus Four Smart MPPT Techniques in Diverse Scenarios" *Inventions* 8, no. 6: 142.
https://doi.org/10.3390/inventions8060142