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

^{1}

^{2}

^{3}

^{4}

^{*}

## 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

- Srinivasarao, P.; Peddakapu, K.; Mohamed, M.; Deepika, K.K.; Sudhakar, K. Simulation and experimental design of adaptive-based maximum power point tracking methods for photovoltaic systems. Comput. Electr. Eng.
**2021**, 89, 106910. [Google Scholar] [CrossRef] - Kishore, D.K.; Mohamed, M.; Sudhakar, K.; Peddakapu, K. Swarm intelligence-based MPPT design for PV systems under diverse partial shading conditions. Energy
**2023**, 265, 126366. [Google Scholar] [CrossRef] - Kishore, D.K.; Mohamed, M.; Sudhakar, K.; Peddakapu, K. An improved grey wolf optimization based MPPT algorithm for photovoltaic systems under diverse partial shading conditions. Proc. J. Phys. Conf. Ser.
**2022**, 2312, 012063. [Google Scholar] [CrossRef] - Syafiq, A.; Pandey, A.; Adzman, N.; Abd Rahim, N. Advances in approaches and methods for self-cleaning of solar photovoltaic panels. Sol. Energy
**2018**, 162, 597–619. [Google Scholar] [CrossRef] - Bourourou, F.; Tadjer, S.A.; Habi, I. Wind Power Conversion Chain Harmonic Compensation using APF Based on FLC. Alger. J. Renew. Energy Sustain. Dev.
**2020**, 2, 75–83. [Google Scholar] [CrossRef] - Rana, K.; Kumar, V.; Sehgal, N.; George, S.; Azar, A.T. Efficient maximum power point tracking in fuel cell using the fractional-order PID controller. In Renewable Energy Systems; Azar, A.T., Kamal, N.A., Eds.; Advances in Nonlinear Dynamics and Chaos (ANDC); Academic Press: Cambridge, MA, USA, 2021; pp. 111–132. [Google Scholar]
- Fekik, A.; Azar, A.T.; Kamal, N.A.; Serrano, F.E.; Hamida, M.L.; Denoun, H.; Yassa, N. Maximum Power Extraction from a Photovoltaic Panel Connected to a Multi-cell Converter. In Proceedings of the International Conference on Advanced Intelligent Systems and Informatics, Cairo, Egypt, 19–21 October 2020; Hassanien, A.E., Slowik, A., Snášel, V., El-Deeb, H., Tolba, F.M., Eds.; Springer International Publishing: Cham, Switzerland, 2021; pp. 873–882. [Google Scholar]
- Ammar, H.H.; Azar, A.T.; Shalaby, R.; Mahmoud, M.I. Metaheuristic Optimization of Fractional Order Incremental Conductance (FO-INC) Maximum Power Point Tracking (MPPT). Complexity
**2019**, 2019, 1–13. [Google Scholar] [CrossRef] - Amara, K.; Malek, A.; Bakir, T.; Fekik, A.; Azar, A.T.; Almustafa, K.M.; Bourennane, E.B.; Hocine, D. Adaptive neuro-fuzzy inference system based maximum power point tracking for stand-alone photovoltaic system. Int. J. Model. Identif. Control
**2019**, 33, 311–321. [Google Scholar] [CrossRef] - Ben Smida, M.; Sakly, A.; Vaidyanathan, S.; Azar, A.T. Control-Based Maximum Power Point Tracking for a Grid-Connected Hybrid Renewable Energy System Optimized by Particle Swarm Optimization. In Advances in System Dynamics and Control; Azar, A.T., Vaidyanathan, S., Eds.; Advances in Systems Analysis, Software Engineering, and High Performance Computing (ASASEHPC); IGI Global: Hershey, PA, USA, 2018; pp. 58–89. [Google Scholar]
- El Aroudi, A. A new approach for accurate prediction of subharmonic oscillation in switching regulators—Part I: Mathematical derivations. IEEE Trans. Power Electron.
**2016**, 32, 5651–5665. [Google Scholar] [CrossRef] - Femia, N.; Petrone, G.; Spagnuolo, G.; Vitelli, M. A technique for improving P&O MPPT performances of double-stage grid-connected photovoltaic systems. IEEE Trans. Ind. Electron.
**2009**, 56, 4473–4482. [Google Scholar] - Pu, Q.; Zhu, X.; Liu, J.; Cai, D.; Fu, G.; Wei, D.; Sun, J.; Zhang, R. Integrated optimal design of speed profile and fuzzy PID controller for train with multifactor consideration. IEEE Access
**2020**, 8, 152146–152160. [Google Scholar] [CrossRef] - Hosseini, S.; Taheri, S.; Pouresmaeil, E.; Espinoza-Trejo, D.R. Enhancement of A Photovoltaic Inverter Efficiency Using A Shade-Tolerant MPPT. In Proceedings of the IECON 2021—47th Annual Conference of the IEEE Industrial Electronics Society, IEEE, Toronto, ON, Canada, 13–16 October 2021; pp. 1–5. [Google Scholar]
- Garcerá, G.; Figueres, E.; Mocholí, A. Novel three-controller average current mode control of DC-DC PWM converters with improved robustness and dynamic response. IEEE Trans. Power Electron.
**2000**, 15, 516–528. [Google Scholar] [CrossRef] - Estcourt, C.; Stirrup, O.; Mapp, F.; Copas, A.; Howarth, A.; Owusus, M.W.; Low, N.; Saunders, J.; Mercer, C.; Flowers, P.; et al. O18. 3 Characteristics and outcomes of people who used Accelerated Partner Therapy for chlamydia in the LUSTRUM cluster cross-over randomised control trial. Sex. Transm. Infect.
**2021**, 97, A58. [Google Scholar] - Bianconi, E.; Calvente, J.; Giral, R.; Mamarelis, E.; Petrone, G.; Ramos-Paja, C.A.; Spagnuolo, G.; Vitelli, M. A fast current-based MPPT technique employing sliding mode control. IEEE Trans. Ind. Electron.
**2012**, 60, 1168–1178. [Google Scholar] [CrossRef] - Hosseini, S.; Taheri, S.; Farzaneh, M.; Taheri, H. A high-performance shade-tolerant MPPT based on current-mode control. IEEE Trans. Power Electron.
**2019**, 34, 10327–10340. [Google Scholar] [CrossRef] - Weidong, J.; Wang, L.; Wang, J.; Zhang, X.; Wang, P. A carrier-based virtual space vector modulation with active neutral-point voltage control for a neutral-point-clamped three-level inverter. IEEE Trans. Ind. Electron.
**2018**, 65, 8687–8696. [Google Scholar] [CrossRef] - Garcia-Teruel, A.; Forehand, D. A review of geometry optimisation of wave energy converters. Renew. Sustain. Energy Rev.
**2021**, 139, 110593. [Google Scholar] [CrossRef] - Guo-Hua, Z.; Bo-Cheng, B.; Jian-Ping, X.; Yan-Yan, J. Dynamical analysis and experimental verification of valley current controlled buck converter. Chin. Phys. B
**2010**, 19, 050509. [Google Scholar] [CrossRef] - El Aroudi, A.; Al-Numay, M.; Garcia, G.; Al Hossani, K.; Al Sayari, N.; Cid-Pastor, A. Analysis of nonlinear dynamics of a quadratic boost converter used for maximum power point tracking in a grid-interlinked PV system. Energies
**2018**, 12, 61. [Google Scholar] [CrossRef][Green Version] - Erickson, R.W.; Maksimović, D. Principles of steady-state converter analysis. In Fundamentals of Power Electronics; Springer: Berlin/Heidelberg, Germany, 2020; pp. 15–41. [Google Scholar]
- El Aroudi, A.; Zhioua, M.; Al-Numay, M.; Garraoui, R.; Al-Hosani, K. Stability Analysis of a Boost Converter with an MPPT Controller for Photovoltaic Applications. In Proceedings of the Mediterranean Conference on Information & Communication Technologies, Saidia, Morocco, 7–9 May 2015; Springer: Berlin/Heidelberg, Germany, 2016; pp. 483–491. [Google Scholar]
- Dongmo Wamba, M.; Montagner, J.P.; Romanowicz, B. Imaging deep-mantle plumbing beneath La Réunion and Comores hot spots: Vertical plume conduits and horizontal ponding zones. Sci. Adv.
**2023**, 9, eade3723. [Google Scholar] [CrossRef] - Wamba, M.; Montagner, J.P.; Romanowicz, B.; Barruol, G. Multi-Mode Waveform Tomography of the Indian Ocean Upper and Mid-Mantle Around the Réunion Hotspot. J. Geophys. Res. Solid Earth
**2021**, 126, e2020JB021490. [Google Scholar] [CrossRef] - Kagho, L.; Dongmo, M.; Pelap, F. Dynamics of an Earthquake under Magma Thrust Strength. J. Earthq.
**2015**, 2015, 1–9. [Google Scholar] [CrossRef][Green Version] - Pelap, F.; Fomethe, A.; Dongmo, M.; Kagho, L.; Tanekou, G.; Makenne, Y. Direction effects of the pulling force on the first order phase transition in a one block model for earthquakes. J. Geophys. Eng.
**2014**, 11, 045007. [Google Scholar] [CrossRef] - Dongmo, M.; Kagho, L.; Pelap, F.; Tanekou, G.; Makenne, Y.; Fomethe, A. Water effects on the first-order transition in a model of earthquakes. Int. Sch. Res. Not.
**2014**, 2014. [Google Scholar] [CrossRef][Green Version] - Giaouris, D.; Elbkosh, A.; Banerjee, S.; Zahawi, B.; Pickert, V. Stability of switching circuits using complete-cycle solution matrices. In Proceedings of the 2006 IEEE International Conference on Industrial Technology, Mumbai, India, 16–18 August 2006; pp. 1954–1959. [Google Scholar]
- Giaouris, D.; Banerjee, S.; Zahawi, B.; Pickert, V. Stability analysis of the continuous-conduction-mode buck converter via Filippov’s method. IEEE Trans. Circuits Syst. I Regul. Pap.
**2008**, 55, 1084–1096. [Google Scholar] [CrossRef][Green Version] - Giaouris, D.; Banerjee, S.; Zahawi, B.; Pickert, V. Control of fast scale bifurcations in power-factor correction converters. IEEE Trans. Circuits Syst. II Express Briefs
**2007**, 54, 805–809. [Google Scholar] [CrossRef][Green Version] - Tucker, W. Computing accurate Poincaré maps. Phys. D Nonlinear Phenom.
**2002**, 171, 127–137. [Google Scholar] [CrossRef] - Giaouris, D.; Banerjee, S.; Stergiopoulos, F.; Papadopoulou, S.; Voutetakis, S.; Zahawi, B.; Pickert, V.; Abusorrah, A.; Al Hindawi, M.; Al-Turki, Y. Foldings and grazings of tori in current controlled interleaved boost converters. Int. J. Circuit Theory Appl.
**2014**, 42, 1080–1091. [Google Scholar] [CrossRef] - Sahan, B.; Vergara, A.N.; Henze, N.; Engler, A.; Zacharias, P. A single-stage PV module integrated converter based on a low-power current-source inverter. IEEE Trans. Ind. Electron.
**2008**, 55, 2602–2609. [Google Scholar] [CrossRef] - Deane, J.H.; Hamill, D.C. Instability, subharmonics and chaos in power electronic systems. In Proceedings of the 20th Annual IEEE Power Electronics Specialists Conference, Milwaukee, WI, USA, 26–29 June 1989; pp. 34–42. [Google Scholar]
- Wolf, A.; Swift, J.B.; Swinney, H.L.; Vastano, J.A. Determining Lyapunov exponents from a time series. Phys. D Nonlinear Phenom.
**1985**, 16, 285–317. [Google Scholar] [CrossRef][Green Version] - Zhang, S.; Deng, Z.; Li, W. A precise Runge–Kutta integration and its application for solving nonlinear dynamical systems. Appl. Math. Comput.
**2007**, 184, 496–502. [Google Scholar] [CrossRef] - Banerjee, S.; Ranjan, P.; Grebogi, C. Bifurcations in two-dimensional piecewise smooth maps-theory and applications in switching circuits. IEEE Trans. Circuits Syst. I Fundam. Theory Appl.
**2000**, 47, 633–643. [Google Scholar] [CrossRef][Green Version] - Robert, B.; Robert, C. Border collision bifurcations in a one-dimensional piecewise smooth map for a PWM current-programmed H-bridge inverter. Int. J. Control
**2002**, 75, 1356–1367. [Google Scholar] [CrossRef] - Wolf, D.M.; Varghese, M.; Sanders, S.R. Bifurcation of power electronic circuits. J. Frankl. Inst.
**1994**, 331, 957–999. [Google Scholar] [CrossRef]

**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 |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## 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