# A Robust Maximum Power Point Tracking Control Method for a PEM Fuel Cell Power System

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## Abstract

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## 1. Introduction

## 2. PEM Fuel Cell

## 3. DC–DC Boost Converter

- The ON state: When the switch S turns ON, the inductor L connects to the DC source voltage. Therefore, the current ${i}_{L}$ moves across the inductor L and the transistor switch s, which results in an increase in the magnitude of ${i}_{L}$ and ${i}_{s}$, while ${V}_{L}$ is approximately equal to the input voltage ${V}_{in}$. On the other hand, during this state, the capacitor C discharges through the load R. The obtained differential equations of the inductor current ${i}_{L}$ and the output voltage ${V}_{out}$ are expressed as follows:$$\left\{\begin{array}{c}\frac{d{i}_{L}}{dt}=\frac{1}{L}\left({V}_{in}\right)\hfill \\ \frac{d{V}_{out}}{dt}=\frac{1}{C}(-{i}_{out})\hfill \end{array}.\right.$$
- The OFF state: When the switch S turns OFF, the inductor L connects to the capacitor C and the load R. Therefore, the current ${i}_{L}$ moves across the inductor L, the diode D, the capacitor C, and the load R, which results in a decrease in the magnitude of ${i}_{L}$ and ${i}_{d}$ (discharging of the inductor L into the load R and the capacitor C). During this state, ${V}_{L}$ is approximately equal to ${V}_{in}-{V}_{out}$. The obtained differential equations of the inductor current ${i}_{L}$ and the output voltage ${V}_{out}$ are expressed as follows:$$\left\{\begin{array}{c}\frac{d{i}_{L}}{dt}=\frac{1}{L}({V}_{in}-{V}_{out})\hfill \\ \frac{d{V}_{out}}{dt}=\frac{1}{C}({i}_{L}-{i}_{out})\hfill \end{array}.\right.$$

## 4. MPPT Control Design

#### 4.1. Current Reference Estimator ${I}_{mpp}$

#### 4.2. Current Regulation

#### 4.2.1. PI Controller

- The first step is to switch off the integral and derivative gains (${K}_{i}=0$ and ${K}_{d}=0$).
- The second step is to increase the ${k}_{p}$ gain from a low/zero value until the first sustained oscillation occurs (Figure 11). The reached gain at the sustained oscillation is noted as a critical value ${k}_{cr}$, while the period of these oscillations is measured as ${P}_{cr}$.
- Finally, taking into account the type of the used controller, ${K}_{p}$ and ${K}_{i}$ can be calculated using the formula given in Table 2.

#### 4.2.2. Backstepping Algorithm

**Step 1.**First, we define the tracking current error as$${e}_{1}={x}_{1}-{I}_{mpp}.$$$$\dot{{e}_{1}}=-(1-{u}_{1})\frac{{x}_{2}}{L}+\frac{{V}_{Stack}}{L}-{\dot{I}}_{ref}$$$${V}_{1}=\frac{1}{2}{e}_{1}^{2}.$$$${\dot{V}}_{1}={e}_{1}{\dot{e}}_{1}={e}_{1}\left(-(1-{u}_{1})\frac{{x}_{2}}{L}+\frac{{V}_{Stack}}{L}-{\dot{I}}_{ref}\right).$$$${\gamma}_{1}=\frac{1}{1-{u}_{1}}\left({b}_{1}{e}_{1}+\frac{{V}_{Stack}}{L}-{\dot{I}}_{ref}\right)$$$${e}_{2}=\frac{{x}_{2}}{L}-{\gamma}_{1}.$$Using Equations (25) and (26), Equation (22) can be written as$${\dot{e}}_{1}=-{b}_{1}{e}_{1}-(1-{u}_{1}){e}_{2}.$$Therefore, the Lyapunov function given in Equation (24) can also be rewritten as$${\dot{V}}_{1}=-{b}_{1}{e}_{1}^{2}-(1-{u}_{1}){e}_{1}{e}_{2}.$$**Step 2.**The aim of this step is to enforce the errors $({e}_{1},{e}_{2})$ to vanish. For this reason, first of all, the dynamics of ${e}_{2}$ must be determined. Using Equations (18), (25) and (27), the time-derivative of ${e}_{2}$ can be obtained as$${\dot{e}}_{2}=-\frac{\dot{{u}_{1}}}{1-{u}_{1}}{\gamma}_{1}+\mathsf{\Psi}$$$$\mathsf{\Psi}=\frac{1}{1-{u}_{1}}\left({b}_{1}^{2}{e}_{1}+(1-{u}_{1}){b}_{1}{e}_{2}-\frac{{\dot{V}}_{Stack}}{L}+{\dot{\dot{I}}}_{ref}\right)+\frac{1}{L}\left(\frac{1-{u}_{1}}{C}{x}_{1}-\frac{{x}_{2}}{RC}\right).$$In order to obtain a stabilizing control law ${u}_{1}$ for the whole system, the following Lyapunov function candidate is proposed:$$V={V}_{1}+\frac{1}{2}{e}_{2}^{2}=\frac{1}{2}{e}_{1}^{2}+\frac{1}{2}{e}_{2}^{2}.$$The time derivative of the above Lyapunov function is obtained by combining Equations (28) and (29):$$\begin{array}{ccc}\hfill \dot{V}& =& {\dot{V}}_{1}+{e}_{2}{\dot{e}}_{2}\hfill \end{array}$$$$\begin{array}{ccc}& =& -{b}_{1}{e}_{1}^{2}+{e}_{2}\left({\dot{e}}_{2}-(1-{u}_{1}){e}_{1}\right).\hfill \end{array}$$It can be easily determined that the global asymptotic stability of the equilibrium $({e}_{1},{e}_{2})=(0,0)$ is achieved only if the time derivative of the error variable ${e}_{2}$ is chosen as$${\dot{e}}_{2}=-{b}_{2}{e}_{2}+(1-{u}_{1}){e}_{1}$$$$\dot{{u}_{1}}=\frac{1-{u}_{1}}{{\gamma}_{1}}\left({b}_{2}{e}_{2}-(1-{u}_{1}){e}_{1}+\mathsf{\Psi}\right).$$

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MPP | Maximum Power Point |

MPPT | Maximum Power Point Tracker |

PEMFCs | proton exchange membrane Fuel Cell systems |

FSC | Fractional Short Circuit |

P&O | Perturb and Observation |

FOC | Fractional Open Circuit |

Inc-Cond | Incremental Conductance |

ESC | Extremum Seeking Control |

SMC | Sliding Mode Control |

CS | Current Sweep |

FLC | Fuzzy Logic Control |

ESC | Eagle Strategy Control |

PSO | Particle Swarm Optimization |

NNC | Neural Network Control |

GA | Genetic Algorithms |

VSIR | Variable Step-size Incremental Resistance algorithm |

RCC | Ripple Correlation Control |

CL | Catalyst Layer |

GDL | Gas Diffusion Layer |

CCM | Continuous Conduction Mode |

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**Figure 5.**Waveforms of different currents and voltages under continuous conduction mode (CCM) operation.

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

Cell operating temperature | T | [K] |

Cell standard temperature | ${T}_{std}$ | 298.15 [K] |

Cell operating current | I | [A] |

Universal constant of the gases | R | 83.143 [J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$K${}^{-1}$] |

Constant of Faraday | F | 96,485.309 [C$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$] |

Maximum current density | ${J}_{max}$ | 0.062 A cm${}^{-1}$ |

Current density | J | [A$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$cm${}^{-2}$] |

Change in the free Gibbs energy | $\Delta G$ | [J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$] |

Change of entropy | $\Delta S$ | [J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$] |

Enthalpy of formation | $\Delta H$ | −285.84 kJ$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$ |

Change in the Gibbs free energy at standard condition | $\Delta {G}^{0}$ | [J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$] |

Change of entropy at standard condition | $\Delta {S}^{0}$ | [J$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$mol${}^{-1}$] |

Electrochemical thermodynamics potential | E | [V] |

Standard potential of the fuel cell | ${E}_{0}$ | 1.229 [V] |

Membrane active area | A | [162 cm${}^{2}$] |

Hydrogen and oxygen partial pressures | ${P}_{{\mathrm{H}}_{2}}$, ${P}_{{\mathrm{O}}_{2}}$ | [atm] |

Oxygen concentration | ${C}_{{\mathrm{O}}_{2}}$ | [mol$\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}$cm${}^{-3}$] |

Fuel cell voltage | ${V}_{fc}$ | [V] |

Activation losses | ${V}_{act}$ | [V] |

Ohmic losses | ${V}_{ohm}$ | [V] |

Concentration losses | ${V}_{conc}$ | [V] |

Constant parameters | $\lambda $ | 0.1 [V] |

Electric charge | Q | [coulombs] |

Equivalent resistance of the electron flow | ${R}_{eq}$ | $\Omega $ |

Proton resistance | ${R}_{p}$ | $\Omega $ |

Parametric coefficients | ${k}_{1}$, ${k}_{2}$, ${k}_{3}$, ${k}_{4}$ | $\frac{{k}_{1}=0.9514\text{}[\mathrm{V}],\text{}{\mathrm{k}}_{2}=-0.00312\text{}[\mathrm{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]}{{k}_{3}=-7.4\times {10}^{-5}\text{}[{\mathrm{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}\mathrm{K}}^{-1}],\text{}{\mathrm{k}}_{4}=1.87\times {10}^{-4}\text{}[\mathrm{V}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{\mathrm{K}}^{-1}]}$ |

Type of Controller | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{T}}_{\mathit{i}}={\mathit{K}}_{\mathit{p}}/{\mathit{K}}_{\mathit{i}}$ | ${\mathit{T}}_{\mathit{d}}={\mathit{K}}_{\mathit{d}}/{\mathit{K}}_{\mathit{p}}$ |
---|---|---|---|

P | ${k}_{cr}$/2 | ∞ | 0 |

PI | ${k}_{cr}$/2.2 | ${P}_{cr}$/1.2 | 0 |

PID | ${k}_{cr}$/1.7 | ${P}_{cr}$/2 | ${P}_{cr}$/8 |

Ideal DC–DC Boost Converter | PI Controller | Backstepping | |||||
---|---|---|---|---|---|---|---|

Parameter | R | L | C | ${K}_{p}$ | ${K}_{i}$ | ${b}_{1}$ | ${b}_{2}$ |

Value | 20 $\Omega $ | $69\times {10}^{-3}$ H | $C=1500\times {10}^{-6}$ F | $0.05$ | $10.7$ | 9 | 220 |

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**MDPI and ACS Style**

Derbeli, M.; Barambones, O.; Sbita, L. A Robust Maximum Power Point Tracking Control Method for a PEM Fuel Cell Power System. *Appl. Sci.* **2018**, *8*, 2449.
https://doi.org/10.3390/app8122449

**AMA Style**

Derbeli M, Barambones O, Sbita L. A Robust Maximum Power Point Tracking Control Method for a PEM Fuel Cell Power System. *Applied Sciences*. 2018; 8(12):2449.
https://doi.org/10.3390/app8122449

**Chicago/Turabian Style**

Derbeli, Mohamed, Oscar Barambones, and Lassaad Sbita. 2018. "A Robust Maximum Power Point Tracking Control Method for a PEM Fuel Cell Power System" *Applied Sciences* 8, no. 12: 2449.
https://doi.org/10.3390/app8122449