# Nonlinear Lyapunov Control of a Photovoltaic Water Pumping System

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. System Component Modelling

#### 2.1. PV Module

**I**) is given by

_{ph}_{SC}stands for the short-circuit current at reference temperature T

_{ref}and denotes the temperature coefficient of short-circuit current [21,22].

#### 2.2. Buck Converter

#### 2.3. DC Motor Model

#### 2.4. Pump

## 3. System Controller

**Step 1:**The insulation level is measured online by using a photo sensor.

**Step 2:**The computing of the reference maximum rotational speed for each insulation level value is done on line. Applying Equation (12), the relation between the solar radiation G and the reference speed ${\omega}_{ref}$ is obtained:

**Step 3:**This step consists in measuring the real rotational speed from the PV generator and comparing it to the reference rotational speed $\left({\omega}_{ref}\right)$.

**Step 4:**In this step, the control variable (duty cycle of the DC-DC converter) is determined so that the actual generated rotational speed from the PV generator will be able to track the reference rotational speed $\left({\omega}_{ref}\right)$ employing the input–output feedback linearization [26] and the Lyapunov stability theory [27].

## 4. Simulation Results

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 3.**The current voltage (I-V) characteristics of the PV generator at three different values of solar insulation (G).

**Figure 4.**The power voltage (P-V) characteristics of the PV generator at three different values of solar insulation (G).

Rated Power | 37 W |

Voltage at maximum power | 16.56 V |

Current at maximum power | 2.25 A |

Open circuit voltage | 21.24 V |

Short circuit current | 2.55 A |

Rated Motor Voltage | 60 V |

Rated motor current | 16.5 A |

Rated voltage speed | $272.3\text{}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ |

Armature resistance | $0.8\text{}\mathsf{\Omega}$ |

Armature inductance | 0.04 H |

Voltage and torque constant | $0.175\text{}\mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s}$ |

Moment of inertia | $0.024\text{}\mathrm{k}\mathrm{g}\text{}{\mathrm{m}}^{2}$ |

Torque of constant for rotational losses | $0.024\text{}\mathrm{N}\text{}\mathrm{m}$ |

Viscose friction coefficient | $001\text{}\mathrm{N}\text{}\mathrm{m}\text{}$ |

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**MDPI and ACS Style**

Jouili, K.; Madani, A.
Nonlinear Lyapunov Control of a Photovoltaic Water Pumping System. *Energies* **2023**, *16*, 2241.
https://doi.org/10.3390/en16052241

**AMA Style**

Jouili K, Madani A.
Nonlinear Lyapunov Control of a Photovoltaic Water Pumping System. *Energies*. 2023; 16(5):2241.
https://doi.org/10.3390/en16052241

**Chicago/Turabian Style**

Jouili, Khalil, and Adel Madani.
2023. "Nonlinear Lyapunov Control of a Photovoltaic Water Pumping System" *Energies* 16, no. 5: 2241.
https://doi.org/10.3390/en16052241