# Bifurcation Analysis of a Photovoltaic Power Source Interfacing a Current-Mode-Controlled Boost Converter with Limited Current Sensor Bandwidth for Maximum Power Point Tracking

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## Abstract

**:**

## 1. Introduction

- Propose a flexible control based on the inductor current filtered by a low-pass filter integrating a relevant MPPT P&O algorithm to estimate the average value of the PV source power;
- Propose suitable orbital stability tools such as Floquet theory to study the stability of the overall system under consideration as a function of the cut-off frequency of the low-pass filter and the amplitude of the ramp signal;
- Develop a bifurcation analysis of the DC-DC power system with MATLAB/SIMULINK based on the fourth-order Runge–Kutta numerical method for a deep study of the stability;
- Develop a bifurcation analysis of the DC-DC power system with the PSIM software that is close to experimental interpretation of the DC-DC system dynamic.

## 2. Materials and Methods

#### 2.1. System Evaluation

#### 2.2. Mathematical Modeling

#### 2.3. Steady-State Analysis

#### 2.4. Floquet Theory

#### 2.4.1. The Piecewise Linear State-Space Switched Model Close to the Maximum Power Point

#### 2.4.2. Stability Analysis Using Floquet Theory

## 3. Results and Discussions

#### 3.1. Floquet Theory on the Stability

#### 3.1.1. Simulations Results in MATLAB/SIMULINK Software

**Remark**

**1.**

**Remark**

**2.**

#### 3.1.2. Simulations Results in PSIM Software

#### 3.2. Bifurcation Behavior from the Nonlinear Circuit-Level Switched Model with the Linear Model of the PV Generator from MATLAB/SIMULINK Software

**Remark**

**3.**

#### 3.3. Bifurcation Behavior from the Nonlinear Circuit-Level Switched Model with the Nonlinear Model of the PV Generator from PSIM Software

**Remark**

**4.**

#### 3.4. Stability Boundaries in the Parameter Space

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Different MPPT control strategies. (

**a**) direct duty cycle control, (

**b**) single-loop voltage mode control, (

**c**) two-loop current mode control with voltage loop closed and (

**d**) current mode control with voltage loop open with a single current sensor for both current and MPPT controls.

**Figure 2.**Boost converter fed by a PV generator with MPPT and current mode controller. (

**a**) the MPPT control is performed by using the PV current. (

**b**) The MPPT control is performed by using the filtered inductor current.

**Figure 4.**The evolution of the current reference ${i}_{ref}$ in terms of the ramp amplitude ${V}_{M}$ according to (9). (

**a**) $E=60$ V. (

**b**) $E=48$ V.

**Figure 5.**MATLAB simulation of evolution of the Floquet multipliers of the PV system by taking the amplitude of the carrier signal amplitude ${V}_{M}$ as a bifurcation parameter for different values of the current sensor bandwidth ${\omega}_{c}$ and DC output voltage E. The critical values of ${V}_{M}$ at which period doubling bifurcation takes place are indicated. (

**a**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=60$ V; (

**b**) ${f}_{c}={f}_{s}$, $E=60$ V; (

**c**) ${f}_{c}=2{f}_{s}$, $E=60$ V; (

**d**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=48$ V; (

**e**) ${f}_{c}={f}_{s}$, $E=48$ V; (

**f**) ${f}_{c}=2{f}_{s}$, $E=48$ V.

**Figure 6.**Bifurcation diagrams in MATLAB software by taking the amplitude of the carrier signal amplitude ${V}_{M}$ as a bifurcation parameter for different values of the current sensor bandwidth ${\omega}_{c}$ and DC output voltage E. (

**a**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=60$ V. (

**b**) ${f}_{c}={f}_{s}$, $E=60$ V. (

**c**) ${f}_{c}=2{f}_{s}$, $E=60$ V. (

**d**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=48$ V. (

**e**) ${f}_{c}={f}_{s}$, $E=48$ V. (

**f**) ${f}_{c}=2{f}_{s}$, $E=48$ V.

**Figure 7.**PSIM software simulation of Evolution of the Floquet multipliers of the PV system by taking the amplitude of the carrier signal amplitude ${V}_{M}$ as a bifurcation parameter for different values of the current sensor bandwidth ${\omega}_{c}$ and DC output voltage E. The critical values of ${V}_{M}$ at which period doubling bifurcation takes place are indicated. (

**a**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=60$ V. (

**b**) ${f}_{c}={f}_{s}$, $E=60$ V. (

**c**) ${f}_{c}=2{f}_{s}$, $E=60$ V. (

**d**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=48$ V. (

**e**) ${f}_{c}={f}_{s}$, $E=48$ V. (

**f**) ${f}_{c}=2{f}_{s}$, $E=48$ V.

**Figure 8.**Lyapunov exponent diagrams in MATLAB/SIMULINK software by taking the amplitude of the carrier signal amplitude ${V}_{M}$ as a bifurcation parameter for different values of the current sensor bandwidth ${\omega}_{c}$ and DC output voltage E. (

**a**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=60$ V. (

**b**) ${f}_{c}={f}_{s}$, $E=60$ V. (

**c**) ${f}_{c}=2{f}_{s}$, $E=60$ V. (

**d**) ${f}_{c}=\frac{{f}_{s}}{2}$, $E=48$ V. (

**e**) ${f}_{c}={f}_{s}$, $E=48$ V. (

**f**) ${f}_{c}=2{f}_{s}$, $E=48$ V.

**Figure 9.**Bifurcation diagrams obtained by taking the amplitude of the carrier signal amplitude ${V}_{M}$ as a bifurcation parameter for different values of the current sensor bandwidth ${\omega}_{c}$ and DC output voltage E. The linearized model of the PV generator close to the MPP was used.

**Figure 10.**Stability boundaries in the plane $E-{V}_{M}$ for different values of cut-off frequency in MATLAB/SIMULINK software. ${f}_{s}$ is fixed at ${f}_{s}=50$ kHz. Where: A(60v,1.42v), B(60v,1.42v), C(60v,1.25v), D(48v,0.93v), E(48v,0.74v), and F(48v,0.65v) are the period splitting type bifurcation occur in the system. A and D are for fc = fs/2; B and E are for fc = fs; and C and E are for fc = 2fs.

**Figure 11.**Stability boundaries for different values of cut-off frequency in PSIM software: (

**a**) in the plane $D-{V}_{M}$; (

**b**) in the plane $E-{V}_{M}$. Where: A(60v,1.42v), B(60v,1.42v), C(60v,1.25v), D(48v,0.93v), E(48v,0.74v), and F(48v,0.65v) are the period splitting type bifurcation occur in the system. A and D are for fc=fs/2; B and E are for fc = fs; and C and E are for fc = 2fs.

Parameters | Values |
---|---|

${C}_{1}$ | 10 $\mu $F |

L | 200 $\mathsf{\mu}$H |

r | 100 m$\mathsf{\Omega}$ |

E | 48 V and 60 V |

${R}_{2}$ | 200 m$\mathsf{\Omega}$ |

${R}_{s}$ | 1 $\mathsf{\Omega}$ |

${C}_{2}$ | 47 $\mathsf{\mu}$F |

${i}_{\mathrm{ref}}$ | updated according to (9) |

${f}_{s}$ | 50 kHz |

${f}_{c}$ | variable |

${V}_{M}$ | Variable |

Parameters | Values |
---|---|

Maximum power ${P}_{\mathrm{mpp}}$ | 85.17 W |

Voltage at maximum power ${V}_{\mathrm{mpp}}$ | 18.28 V |

Current at maximum power ${V}_{\mathrm{mpp}}$ | 4.66 A |

Maximum power ${P}_{max}$ | 85.17 W |

Short-circuit current ${I}_{sc}$ | 5 A |

Open-circuit voltage ${V}_{oc}$ | $21.1$ V |

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## Share and Cite

**MDPI and ACS Style**

Kengne, E.R.M.; Kammogne, A.S.T.; Siewe, M.S.; Tamo, T.T.; Azar, A.T.; Mahlous, A.R.; Tounsi, M.; Khan, Z.I.
Bifurcation Analysis of a Photovoltaic Power Source Interfacing a Current-Mode-Controlled Boost Converter with Limited Current Sensor Bandwidth for Maximum Power Point Tracking. *Sustainability* **2023**, *15*, 6097.
https://doi.org/10.3390/su15076097

**AMA Style**

Kengne ERM, Kammogne AST, Siewe MS, Tamo TT, Azar AT, Mahlous AR, Tounsi M, Khan ZI.
Bifurcation Analysis of a Photovoltaic Power Source Interfacing a Current-Mode-Controlled Boost Converter with Limited Current Sensor Bandwidth for Maximum Power Point Tracking. *Sustainability*. 2023; 15(7):6097.
https://doi.org/10.3390/su15076097

**Chicago/Turabian Style**

Kengne, Edwige Raissa Mache, Alain Soup Tewa Kammogne, Martin Siewe Siewe, Thomas Tatietse Tamo, Ahmad Taher Azar, Ahmed Redha Mahlous, Mohamed Tounsi, and Zafar Iqbal Khan.
2023. "Bifurcation Analysis of a Photovoltaic Power Source Interfacing a Current-Mode-Controlled Boost Converter with Limited Current Sensor Bandwidth for Maximum Power Point Tracking" *Sustainability* 15, no. 7: 6097.
https://doi.org/10.3390/su15076097