**1. Introduction**

Today, protecting our planet is a major issue that involves several policies pertaining to transport and energy. Respecting the Kyoto Protocol and Paris Agreement on the reduction of greenhouse gas emissions, and keeping a global temperature rise this century below two degrees Celsius, requires drastic measures in favour of energy savings and the development of renewable energy. Indeed, given the increase in the global population, attention is being paid to the fact that energy supplies are necessarily limited, and that the risk of one day being faced with an energy shortage may become a reality. The transport sector today is seriously threatened because it is, on the one hand, extremely dependent on oil and, on the other hand, it is partly responsible for greenhouse gas emissions. In this respect, the use of fuel cells (FCs) in a traction system of electric vehicles is a hopeful solution, because it ultimately promises zero pollution [1]. In addition, the hydrogen sector has the advantage of being able to reduce the dependence of the transport sector on fossil fuels. Fuel cell electric vehicles (FCEVs) are classified as zero-emission vehicles (ZEVs) because they only release water. Therefore, hydrogen fuel cells have been targeted for their potential to contribute to decarbonization in the transportation sector [2,3]. The first FCEVs, which use polymer electrolyte membrane fuel cells (PEMFCs), were introduced in 2013 [4,5]. The advantages of these vehicles relative to current battery electric vehicles (BEVs) include higher driving ranges (over 500 km) and faster refuelling (3–5 min to refill the hydrogen storage tank).Therefore, the PEMFC is the essential choice for developing distributed generation power systems, hybrid electric vehicles, and other emerging fuel

**Citation:** Belhaj, F.Z.; El Fadil, H.; El Idrissi, Z.; Intidam, A.; Koundi, M.; Giri, F. New Equivalent Electrical Model of a Fuel Cell and Comparative Study of Several Existing Models with Experimental Data from the PEMFC Nexa 1200 W. *Micromachines* **2021**, *12*, 1047. https://doi.org/10.3390/mi12091047

Academic Editor: Francisco J. Perez-Pinal

Received: 15 May 2021 Accepted: 21 July 2021 Published: 30 August 2021

**Publisher's Note:** MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations.

**Copyright:** © 2021 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/).

cell applications. It is therefore important for electrical and automation engineers and researchers to understand the dynamic behaviour of the PEM fuel cell for its successful use in different applications. In the literature, many research works are attempts to develop models of the PEM fuel cell. trical and automation engineers and researchers to understand the dynamic behaviour of the PEM fuel cell for its successful use in different applications. In the literature, many research works are attempts to develop models of the PEM fuel cell. In the beginning, electrochemistry-based models of the PEM fuel cell were introduced [6,7]. Then, dynamic models started to emerge [8–12]. In [13] a dynamic model of

is the essential choice for developing distributed generation power systems, hybrid electric vehicles, and other emerging fuel cell applications. It is therefore important for elec-

*Micromachines* **2021**, *12*, x 2 of 15

In the beginning, electrochemistry-based models of the PEM fuel cell were introduced [6,7]. Then, dynamic models started to emerge [8–12]. In [13] a dynamic model of PEMFC, using the exact linearization approach, was presented. Although these models provide a certain understanding of the PEMFC, they remain insufficient to design adequate controllers for PEM fuel cell systems. It is for this reason that state-space models were introduced in some works [14,15]. However, state-space models are further complicated because they are highly nonlinear, and involve a large number of state variables and parameters. Then, some works have attempted to develop equivalent electric models, because they are still simple and easy to understand and to implement [16–25]. PEMFC, using the exact linearization approach, was presented. Although these models provide a certain understanding of the PEMFC, they remain insufficient to design adequate controllers for PEM fuel cell systems. It is for this reason that state-space models were introduced in some works [14,15]. However, state-space models are further complicated because they are highly nonlinear, and involve a large number of state variables and parameters. Then, some works have attempted to develop equivalent electric models, because they are still simple and easy to understand and to implement [16–25] The present work investigates different classes of models proposed in the literature.

The present work investigates different classes of models proposed in the literature. More specifically, three models are presented: a nonlinear state-space model, a generic dynamic model integrated into MATLAB/Simulink, and an equivalent electric RC circuit. Using the dynamic behaviour of a 1200-W commercialized PEMFC, a new equivalent electric model is proposed. A comparative study between the proposed model and the previous models is conducted, showing the pros and cons of each model. More specifically, three models are presented: a nonlinear state-space model, a generic dynamic model integrated into MATLAB/Simulink, and an equivalent electric RC circuit. Using the dynamic behaviour of a 1200-W commercialized PEMFC, a new equivalent electric model is proposed. A comparative study between the proposed model and the previous models is conducted, showing the pros and cons of each model. The rest of the paper is organized as follows: In Section 2, the electrochemical prin-

The rest of the paper is organized as follows: In Section 2, the electrochemical principle of a PEMFC is presented. Section 2.1 is devoted to the presentation of the nonlinear statespace model. A generic dynamical model integrated into MATLAB/Simulink is illustrated in Section 2.2. In Section 2.3, an equivalent electrical RC circuit is presented. The proposed equivalent electrical RL circuit is shown in Section 2.4. Section 3. is devoted to the experimental behaviour of a 1200-W commercialized PEMFC. The comparative study between different models is conducted in Section 4. A conclusion and a reference list end the paper. ciple of a PEMFC is presented. Section 2.1 is devoted to the presentation of the nonlinear state-space model. A generic dynamical model integrated into MATLAB/Simulink is illustrated in Section 2.2. In Section 2.3, an equivalent electrical RC circuit is presented. The proposed equivalent electrical RL circuit is shown in Section 2.4. Section 3. is devoted to the experimental behaviour of a 1200-W commercialized PEMFC. The comparative study between different models is conducted in Section 4. A conclusion and a reference list end the paper.

## **2. Theoretical Principle 2. Theoretical Principle** A fuel cell (FC) is an electrochemical energy generator used to directly transform the

A fuel cell (FC) is an electrochemical energy generator used to directly transform the chemical energy of a fuel (hydrogen, hydrocarbons, alcohols, etc.) into electrical energy. Figure 1 shows a schematic of a hydrogen PEMFC. The FC core consists of three elements, including two electrodes—an oxidizing anode (electron emitter), and a reducing cathode (electron collector)—separated by an electrolyte. The FC is supplied by an injection of hydrogen at the anode and air at the cathode. Continuous electrical energy is then available across the FC. chemical energy of a fuel (hydrogen, hydrocarbons, alcohols, etc.) into electrical energy. Figure 1 shows a schematic of a hydrogen PEMFC. The FC core consists of three elements, including two electrodes—an oxidizing anode (electron emitter), and a reducing cathode (electron collector)—separated by an electrolyte. The FC is supplied by an injection of hydrogen at the anode and air at the cathode. Continuous electrical energy is then available across the FC.

**Figure 1.** Schematic diagram of a PEMFC. **Figure 1.** Schematic diagram of a PEMFC.

In the core of a hydrogen fuel cell of the PEMFC type, two electrochemical reactions occur successively [26,27]:

At the anode: catalytic oxidation of the hydrogen, which dissociates from its electrons: \_

In the core of a hydrogen fuel cell of the PEMFC type, two electrochemical reactions

At the anode: catalytic oxidation of the hydrogen, which dissociates from its elec-

$$H\_2 \rightarrow 2H^+ + 2e^- \tag{1}$$

ions that

At the cathode: catalytic reduction of the oxygen, which captures the *H*<sup>+</sup> ions that have passed through the electrolyte membrane, as well as the electrons arriving from the external circuit. The reaction produces heat and water: have passed through the electrolyte membrane, as well as the electrons arriving from the external circuit. The reaction produces heat and water: 1 ܪ2 + <sup>ଶ</sup>0 ା +2݁ (2) (ݐ݁ܽℎ(ܳ + ܱଶܪ → \_

At the cathode: catalytic reduction of the oxygen, which captures the *H*<sup>+</sup>

$$\frac{1}{2}O\_2 + 2H^+ + 2e^- \rightarrow H\_2O + Q(heat) \tag{2}$$

To evaluate the PEMFC's performance, and for control purposes, several mathematical models of PEMFC have been developed in the literature. They can be classified into three main categories: into three main categories: Static models representing the input–output behaviour of the FC—in particular, the nonlinear current–voltage characteristic (see Figure 2). The output voltage of the fuel cell

Static models representing the input–output behaviour of the FC—in particular, the nonlinear current–voltage characteristic (see Figure 2). The output voltage of the fuel cell is dependent on the thermodynamically predicted fuel cell voltage output, and three major losses: activation losses (due to the electrochemical reaction), ohmic losses (due to the ionic electronic condition), and concentration losses (due to mass transport). is dependent on the thermodynamically predicted fuel cell voltage output, and three major losses: activation losses (due to the electrochemical reaction), ohmic losses (due to the ionic electronic condition), and concentration losses (due to mass transport). Nonlinear state-space representing the internal behaviour of the fuel cell and equivalent electrical circuits.

*Micromachines* **2021**, *12*, x 3 of 15

2

occur successively [26,27]:

trons:

**Figure 2.** Nonlinear *i*–*v* characteristic of the fuel cell. **Figure 2.** Nonlinear*i*–*v* characteristic of the fuel cell.

In this paper a comparison between four models is investigated: a nonlinear state-space model, a generic model from MATLAB Toolbox, an equivalent electrical cir-Nonlinear state-space representing the internal behaviour of the fuel cell and equivalent electrical circuits.

cuit RC, and a new proposed equivalent electrical circuit RL. The resulting models will be compared to the experimental results using a 1.2-kW fuel cell module from Ballard (the Nexa 1200). In this paper a comparison between four models is investigated: a nonlinear statespace model, a generic model from MATLAB Toolbox, an equivalent electrical circuit RC, and a new proposed equivalent electrical circuit RL. The resulting models will be compared to the experimental results using a 1.2-kW fuel cell module from Ballard (the Nexa 1200).

### *2.1. Nonlinear State-Space Model of the PEMFC (NLM) 2.1. Nonlinear State-Space Model of the PEMFC (NLM)*

It has been suggested in many studies [12,15,16,28] that a nonlinear state-space model of the PEM fuel cell could be represented by Equations (3)–(8). In this model, the open-circuit output voltage of the PEM fuel cell, mass balance and thermodynamic energy balance, irreversible voltage losses, and the formation of the charged double layer in the PEM fuel cell is modelled: It has been suggested in many studies [12,15,16,28] that a nonlinear state-space model of the PEM fuel cell could be represented by Equations (3)–(8). In this model, the opencircuit output voltage of the PEM fuel cell, mass balance and thermodynamic energy balance, irreversible voltage losses, and the formation of the charged double layer in the PEM fuel cell is modelled:

$$\dot{\mathbf{x}}\_1 = -\theta\_1 \mathbf{x}\_1 + \theta\_1 \mathbf{u}\_{T\_R} - L(\mathbf{x}) I\_{f\mathbf{c}} \tag{3}$$

$$\dot{\mathbf{x}}\_2 = 2\theta\_2 \mathbf{x}\_1 \boldsymbol{\mu}\_{P\_A} - 2\theta\_2 \mathbf{x}\_1 \mathbf{x}\_2 - \theta\_3 \mathbf{x}\_1 \mathbf{I}\_{f\mathbf{c}} \tag{4}$$

$$\dot{\mathbf{x}}\_3 = 2\theta\_4 \mathbf{x}\_1 \boldsymbol{\mu}\_{P\_\mathbb{C}} - 2\theta\_4 \mathbf{x}\_1 \mathbf{x}\_3 - \theta\_5 \mathbf{x}\_1 I\_{fc} \tag{5}$$

$$
\dot{\mathbf{x}}\_4 = 2\theta\_6 \mathbf{x}\_4 \mathbf{x}\_1 + 2\theta\_5 \mathbf{x}\_1 I\_{fc} \tag{6}
$$

$$
\dot{\mathfrak{x}}\_{\mathfrak{5}} = -\theta\_{\mathfrak{5}}\mathfrak{x}\_{\mathfrak{5}} + \theta\_{\mathfrak{6}}I\_{f\mathfrak{c}} \tag{7}
$$

where *x*<sup>1</sup> = *T* is a stack temperature; *x*<sup>2</sup> = *PH*<sup>2</sup> is the partial pressure of hydrogen; *x*<sup>3</sup> = *PO*<sup>2</sup> is the partial pressure of oxygen; *x*<sup>4</sup> = *PH*2*<sup>O</sup>* is the partial pressure of water; *x*<sup>5</sup> = *Vf c* is the output voltage of the PEM fuel cell; *If c* is the stack current; *uP<sup>A</sup>* is the channel pressure of hydrogen; *uP<sup>C</sup>* is the channel pressure of oxygen; *uT<sup>R</sup>* is room temperature; and the involved parameters and functions are given as follows:

$$L(\mathbf{x}) = n\_s \left[ \left( \frac{2F\_0^{\text{eff}}}{M\_{fc}C\_{fc}} \right) + \left( \frac{Rx\_1}{FM\_{fc}C\_{fc}} \right) \ln \left( \frac{x\_2 \alpha\_0^{\text{eff}}}{x\_4} \right) - V^{Act} - V^{Cont} - V^O \right]$$

$$\theta\_1 = \frac{h\_3 n\_s A\_3}{M\_{fc}C\_{fc}}$$

$$\theta\_2 = \left[ \frac{\left( \frac{R \left( m\_{H\_2O} \right)^a\_{in} x\_1}{A\_3} \right)}{\left( V\_a \left( R\_{H\_2O} \right)^a\_{in} \right)} \right]$$

$$\theta\_3 = \left[ \frac{R \pi\_1}{4V\_c \bar{F}} \right]$$

$$\theta\_4 = \left[ \frac{\left( R \left( m\_{H\_2O} \right)^c\_{in} x\_1 \right)}{\left( V\_c \left( R\_{H\_2O} \right)^c\_{in} \right)} \right]$$

$$\theta\_5 = \left[ \frac{R \pi\_1}{4V\_c \bar{F}} \right]$$

$$\theta\_6 = \left[ \frac{\left( R \left( m\_{H\_2O} \right)^c\_{in} \left( P\_{H\_2O}^H - \mathbf{x}\_4 \right) \right)}{\left( V\_c \left( R\_{H\_2O} \right)^c\_{in} \right)} \right]$$

$$\theta\_7 = \frac{\Gamma \left( R\_{AC} + R\_{C0} \right)}{\Gamma \left( R\_{AC} + R\_{C0} \right)} \tag{8}$$

$$\theta\_8 = \frac{1}{\bar{C}} \tag{9}$$
