**1. Introduction**

Due to its abundance in the universe, hydrogen has become one of the most important fuels for energy production. Hydrogen represents up to more than 75% of all normal matter mass, and it accounts for over 90% of all atoms on earth [1]; it could be produced by either simple methods, such as the electrolysis of water, or industrial methods using steam reforming. The production cost of hydrogen is expected to fall by 50% by the middle of this century, and that could pave the way for more sustainable sources of energy [2]. The latter has encouraged thousands of scientists and researchers to pursue research in hydrogen cells.

A proton exchange membrane fuel cell (PEMFC), which uses hydrogen as the main fuel, has recently attracted great attention due to its cleanliness, high efficiency, high power density and quiet operation [3]. It can be used for a wide range of applications, including automotive, stationary and portable power supplies [4–7]. For most of those applications, the PEMFC is usually used in conjunction with a DC–DC power converter that generates highly regulated DC voltage for end-use. Therefore, the control design plays the main role in a PEMFC power system, not only for performance improvement reasons but also for safety operation.

During the last few years, many control algorithms have been designed for PEMFC power systems; the pros and cons of the recently reported ones are listed in Table 1. Hence, linear proportional integral (PI), proportional derivative (PD) and proportional integral derivative (PID) have been, respectively, used by various research groups/researchers [8–10], to keep the PEMFC operating at an appropriate power point. Although these controllers are especially sensitive when they face a large load variation, results showed a gradual

**Citation:** Derbeli, M.; Charaabi, A.; Barambones, O.; Napole, C. High-Performance Tracking for Proton Exchange Membrane Fuel Cell System PEMFC Using Model Predictive Control. *Mathematics* **2021**, *9*, 1158. https://doi.org/10.3390/ math9111158

Academic Editors: Mikhail Posypkin, Andrey Gorshenin and Vladimir Titarev

Received: 19 April 2021 Accepted: 18 May 2021 Published: 21 May 2021

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and smooth rise to the desired operating power point with an acceptable tracking performance. To increase the robustness of the PID and obtain a better dynamic performance, various research groups/researchers [11] have applied a fractional order proportional integral derivative (FOPID) controller to a DC–DC four-switch buck-boost (FSBB) converter used in a PEMFC power system. The obtained results have shown that the proposed method achieved better performance in comparison with the integer-order and Two-Zero/Three-Pole (TZTP) controller. Hence, an overall efficiency of 92%, more than the one obtained with TZTP, can be retained using the FOPID. The performances of the PID have also been improved by various research groups/researchers [12] via the application of the slap swarm algorithm (PID-SSA). Comparative results with other methods, such as incremental resistance algorithm (IRA), mine-blast algorithm (MBA), and grey wolf optimizer (GWM), have indicated better performance of the proposed PID-SSA in terms of efficiency and reliability. However, despite the massive work done on improving the performance of the PID, it is still sensitive to cope with the non-linearity of the power converter, which leads many researchers to focus on the non-linear algorithms.

Various research groups/researchers [13] have proposed fuzzy logic control (FLC) to overcome the drawbacks of the conventional P&O, where the results have indicated a chattering reduction of 78.6% and an improvement of 63% in the settling time. To improve the performance of the FLC, various research groups/researchers [14] have proposed particle swarm optimization (FLC-PSO). Comparative results with the FLC have demonstrated the effectiveness of the FLC-PSO in reducing the overshoot from 65.833% to 63.115% while ensuring high tracking efficiency (99.39%). However, despite the reduction of 2%, an overshoot up to more than 63% is still undesirable. Reddy and Sudhakar [15] optimized the FLC via an adaptive neuro-fuzzy inference system (ANFIS). Simulation and experimental results have indicated that an increase of 1.95% in the average DC link and a reduction of 17.74% in the average time taken to reach the operating power point can be achieved using the proposed ANFIS algorithm.

The artificial neural networks and meta-heuristic algorithms have also been used by various research groups/researchers [16–19]. Hence, in comparison with FLC, efficiency improvements and a faster response of 45% are obtained by various research groups/researchers [16] via the application of the neural network algorithm (NNA). The latter was also proposed by [17] to overcome the drawbacks of the P&O. The obtained results showed that a reduction of 86% and 74%, respectively, in power oscillations and settling time can be achieved. In [18], a genetic algorithm (GA) was used to improve the power quality of the PV generator. Results have demonstrated that in comparison with the conventional P&O and the incremental conductance (IC), the proposed GA can achieve a reduction of 97% in the oscillations of output power. Khanam et al. [19] made a comparative study among ant colony optimization (ACO), particle swarm optimization (PSO), differential evolution (DE) and P&O. Results have demonstrated the effectiveness of the ACO in terms of convergence time over the other proposed methods. Hence, in comparison with P&O, a reduction of 90.61% and 5.13% are, respectively, obtained via the application of ACO and PSO.

The application of the sliding mode control (SMC) for the PEMFC system was proposed by various research groups/researchers [3,20,21]. To counteract the chattering phenomenon of the SMC, integral fast terminal sliding mode control (IFTSMC), backstepping sliding mode control (BSMC), high-order sliding mode based on twisting (TA), super-twisting (STA), prescribed convergence law (PCL) and quasi continue (QC) have been, respectively, proposed by [21–26]. Results have demonstrated that high chattering reductions such as 84% and 91% via the application of the QC and STA can be achieved using the proposed algorithms.


**Table 1.** Summary of the recently reported approaches used for the PEMFC power system.

Due to their significant benefits, predictive control methods have attracted the intention of many researches and they have been implemented in a wide range of applications, including power converters, actuator faults, pharmaceuticals industry, chemical processes, and induction motors [27–38]. Hence, in comparison with the conventional P&O algorithm, an improvement of 10.52% in the overall PV system efficiency was achieved by various research groups/researchers [27] via the application of the MPC technique.

In [28], an overall efficiency of 90% for a grid connected system was achieved by applying the MPC for a three-phase inverter, where the efficiency was approximately 98% for the maximum power point tracking (MPPT) control method and 92% for the inverter. A Lyapunov-function-based MPC was proposed by authors of [29], where the results showed that the proposed control strategy maintains the active and reactive powers close to the desired values with an error of less than 3%. Various research groups/researchers [30] have proposed a combination of MPC with an extended Kalman filter (EKF) for a two-level inverter. High performances in terms of robustness and potential noise rejection were obtained. Successful MPP tracking with an efficiency of up to 98% was obtained by various research groups/researchers [31]. In the latter, the MPC is proposed for a boost converter used in a renewable energy system. Various research groups/researchers [32] have compared the MPC with different algorithms, such as IncCond, hill climbing, PSO, and FLC. Except for the design complexity, results have demonstrated that the proposed MPC has succeeded over the other methods in terms of efficiency, steady-state oscillation, tracking speed and accuracy.

In this work, an MPC based on a global minimum cost function for a two-step horizon was designed and implemented in a boost converter integrated with a Heliocentric hy-ExpertTM fuel cell FC-50W. The aim is to maintain the system operating point at an adequate and efficient power stage with high-performance tracking. First, the experimental system, including the fuel cell, the dSPACE, the converter and the programmable load, is explained. Then, the proposed method is designed for a two-step horizon, wherein the cost function is adopted based on the stack current. For investigation, the effectiveness of the proposed method is revealed through a comparison study with a PI controller, which is tuned through the Ziegler–Nichols technique. Finally, some conclusions and perspectives are pointed out.
