# PEMFC Current Control Using a Novel Compound Controller Enhanced by the Black Widow Algorithm: A Comprehensive Simulation Study

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## Abstract

**:**

## 1. Introduction

#### 1.1. Motivations

#### 1.2. State of the Art

#### 1.3. Contributions

#### 1.4. Structure Overview

## 2. PEM Fuel Cell Power System Modeling

#### 2.1. Nernst Voltage

#### 2.2. Activation Polarization

#### 2.3. Ohmic Polarization

#### 2.4. Concentration Polarization

## 3. DC/DC Boost Converter Linked to PEMFC

- $\left({S}_{1}\right)$ is ON: The inductor current will increase linearly until it reaches a peak value of current and, at this point, the voltage around the inductor will be equal to the input voltage source: ${V}_{L}$ = ${V}_{s}$. In this step, the current in the inductor ${i}_{L}$ and the output voltage ${V}_{o}$ are dependent on the following dynamic (13):$$\left\{\begin{array}{c}\frac{d{i}_{L}}{dt}=\frac{1}{L}\left({V}_{s}\right)\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & \\ \frac{d{V}_{o}}{dt}=\frac{1}{RC}(-{V}_{o})\hfill \end{array}\right.$$
- $\left({S}_{1}\right)$ is OFF: In this case, the inductor current ${i}_{L}$ gets discharged into capacity C; at the end of this action, the voltage in the inductor will be ${V}_{L}$ = ${V}_{s}$−${V}_{o}$. At this point, the system in (13) shifts into the following (i.e., (14)):$$\left\{\begin{array}{c}\hfill \frac{d{i}_{L}}{dt}=\frac{1}{L}({V}_{s}-{V}_{o})\hfill \\ \hfill \phantom{\rule{1.em}{0ex}}\hfill & \\ \hfill \frac{d{V}_{o}}{dt}=\frac{1}{C}({i}_{L}-{i}_{o})\hfill \end{array}\right.$$

## 4. Control Design

#### 4.1. Reference Current Based $P\&O$ MPPT

#### 4.2. Super-Twisting Algorithm

#### 4.3. Novel PI Sliding Mode Super-Twisting Controller

#### 4.4. Stability Proof of STA and PISMCSTA

## 5. Optimization Using the Black Widow Algorithm

#### 5.1. Initializing the Population

#### 5.2. Reproduction

#### 5.3. Cannibalism

#### 5.4. Mutation

#### 5.5. Update Population

#### 5.6. Stop Conditions

## 6. Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PI | Proportional–integral |

SMC | Sliding mode control |

PISMCSTA | PI sliding mode controller-based super-twisting algorithm |

PEMFCs | Proton exchange membrane fuel cells |

SGDM | Stochastic gradient descent with momentum |

HOSM-TA | High-order sliding mode-based twisting algorithm |

PID | Proportional–integral–derivative |

FLC | Fuzzy logic controller |

IFLC | Incremental fuzzy logic controller |

AFLC | Adaptive fuzzy controller |

NN | Neural network |

NNFF | Neural network feed-forward |

BP-NN | Back propagation neural network |

ISE | Integral square error |

IAE | Integral absolute error |

ITAE | Integral time-weighted absolute error |

PWM | Pulse-width modulation |

BWOA | Black widow optimization algorithm |

CR | Cannibalism rate |

MR | Mutation rate |

PR | Procreating rate |

MPPT | Maximum power point-tracking |

$P\&O$ | Perturb and observe |

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**Figure 6.**The BWOA flowchart [46].

**Figure 9.**PEMFC polarization curves; (

**a**) P–I curve under different temperature and fixed pressure; (

**b**) V–I curve under different temperature and fixed pressure; (

**c**) P–I curve under different pressure and fixed temperature; (

**d**) V–I curve under different pressure and fixed temperature.

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

A | 162 cm^{2} |

$\gamma $ | 23 |

l | 175 × 10${}^{-6}$ cm |

$\psi $ | 0.1 V |

${R}_{C}$ | 0.0003 |

${J}_{M}$ | 0.062 A · cm^{−1} |

${N}_{Cells}$ | 10 |

${\zeta}_{1}$ | − 0.9514 V |

${\zeta}_{2}$ | −0.00312 V/K |

${\zeta}_{3}$ | −7.4 × 10${}^{-5}$ V/K |

${\zeta}_{4}$ | 1.87 × 10${}^{-4}$ V/K |

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

Inductance | 69 mH |

Capacitor | $1.5$ mF |

Maximum switching frequency | 10 kHz |

Maximum input voltage | 25 V |

Maximum input current | 15 A |

Maximum output voltage | 80 V |

Maximum output current | 2 A |

Algorithm | Range | $\mathit{\lambda}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{K}}_{3}$ |
---|---|---|---|---|---|---|---|

BWOA | Min | $fixed=0.5$ | 0 | 0 | 0 | 0 | 0 |

Max | $fixed=0.5$ | 1 | 1 | 10 | 10 | 10 |

Controller | $\mathit{\lambda}$ | ${\mathit{K}}_{\mathit{p}}$ | ${\mathit{K}}_{\mathit{i}}$ | ${\mathit{K}}_{1}$ | ${\mathit{K}}_{2}$ | ${\mathit{K}}_{3}$ |
---|---|---|---|---|---|---|

STA | $0.5$ | 0 | 0 | $0.2610$ | $6.5047$ | − |

PISMCSTA | $0.5$ | $0.0172$ | $0.5049$ | $0.1148$ | $0.2642$ | $7.8458$ |

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## Share and Cite

**MDPI and ACS Style**

Silaa, M.Y.; Barambones, O.; Cortajarena, J.A.; Alkorta, P.; Bencherif, A.
PEMFC Current Control Using a Novel Compound Controller Enhanced by the Black Widow Algorithm: A Comprehensive Simulation Study. *Sustainability* **2023**, *15*, 13823.
https://doi.org/10.3390/su151813823

**AMA Style**

Silaa MY, Barambones O, Cortajarena JA, Alkorta P, Bencherif A.
PEMFC Current Control Using a Novel Compound Controller Enhanced by the Black Widow Algorithm: A Comprehensive Simulation Study. *Sustainability*. 2023; 15(18):13823.
https://doi.org/10.3390/su151813823

**Chicago/Turabian Style**

Silaa, Mohammed Yousri, Oscar Barambones, José Antonio Cortajarena, Patxi Alkorta, and Aissa Bencherif.
2023. "PEMFC Current Control Using a Novel Compound Controller Enhanced by the Black Widow Algorithm: A Comprehensive Simulation Study" *Sustainability* 15, no. 18: 13823.
https://doi.org/10.3390/su151813823