# PEM Fuel Cell Emulators: A Review

Ángel Hernández-Góm, Diego Langari-Cordoba, Panfilo Martinez-Rodriguez, Damien Guilbert, Victor Ramirez, Belem Saldivar

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Ángel Hernández-Góm, Diego Langari-Cordoba, Panfilo Martinez-Rodriguez, Damien Guilbert, Victor Ramirez, et al.. PEM Fuel Cell Emulators: A Review. Electronics, 2023, 12 (13), pp.2812. 10.3390/electronics12132812. hal-04140410

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Review

# PEM Fuel Cell Emulators: A Review

Ángel Hernández-Gómez<fnref n="1" />,*, Diego Langarica-Cordoba <fnref n="1" />, Panfilo R. Martinez-Rodriguez <fnref n="1" />, Damien Guilbert <fnref n="2" />,*, Víctor Ramírez <fnref n="3" />,*, and Belem Saldivar <fnref n="5" />

<figure><img src="image_2.png" /></figure>

check for
updates

**Citation:** Hernández-Gómez, A.; Langarica-Cordoba, D.; Martinez-Rodriguez, P.; Guilbert, D.; Ramirez, V.; Saldivar B. PEM Fuel Cell Emulators: A Review. *Electronics* 2023, 12, 2812. https://doi.org/10.3390/electronics12132812

Academic Editor: Fabio Corti

Received: 28 May 2023

Revised: 20 June 2023

Accepted: 22 June 2023

Published: 25 June 2023

<figure><img src="image_3.png" /></figure>

**Copyright:** © 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/).

**Abstract:** Proton exchange membrane fuel cell (PEMFC) emulators are feasible solutions for conducting low-cost and safe developments. These types of systems have attracted the attention of global PEMFC manufacturers and research groups over the last few years. Owing to these emulators, it has been possible to develop and optimize PEMFC systems including power electronics and control without the need to use or damage a real PEMFC. However, despite the importance of PEMFC emulators in research, reported studies on this topic remain scarce. For this reason, this review describes the main characteristics and different types of PEMFC emulators (i.e., pseudo and electronic emulators), providing a basis for new emulator prototypes. Additionally, in this paper, the mathematical models that complement PEMFC emulators are presented (i.e., these models and emulators generate reliable measurements compared with real PEMFC systems). Examples of electronic circuit designs based on mathematical models (electrical and heat) are also presented to give some insight into the construction of new PEMFC emulators. Therefore, this paper proposes tools for the construction of new PEMFC emulators to boost the development of this technology.

**Keywords:** circuit electronic modeling; current ripple; DC-DC converter; hardware in a loop; PEMFC; PEMFC emulators

## 1. Introduction

Due to the high costs of fossil fuels and growing concerns about greenhouse gas emissions, researchers are looking for renewable energy sources, with the aim of sustainability with accessible cost, high efficiency, and low-environmental-impact power conversion. Therefore, the interest in power-generating systems such as photovoltaics, wind turbines, and fuel cells (FCs) has increased. However, the intermittency and instability of renewable energies, such as solar and wind, have produced challenges for the stable operation of electrical systems, creating temporal and spatial gaps between energy consumption by end users and energy availability. Therefore, additional energy storage technology is needed as an effective means to help achieve stable and efficient renewable energy operation [1]. FC systems are not subject to intermittent restrictions, making them safer and more reliable. Additionally, FC systems have been shown to be sources of clean energy, environmentally friendly, and sustainable due to their higher energy efficiency, reduced emissions, and high energy density [2,3].

The primary function of FCs is to convert chemical energy from gaseous fuel into electricity. FCs can also serve as substitute stationary and mobile power sources [4].

<footnote n="1">¹ School of Sciences, Universidad Autónoma de San Luis Potosí (UASLP), San Luis Potosí 78295, Mexico; angel-hernandez-1@hotmail.com (Á.H.-G.); diego.langarica@uaslp.mx (D.L.-C.); panfilo.martinez@uaslp.mx (P.R.M.-R.)</footnote>

<footnote n="2">² Group of Research in Electrical Engineering of Nancy (GREEN), Université de Lorraine, F-54000 Nancy, France</footnote>

<footnote n="3">³ Department of renewable energy, Centro de Investigación Científica de Yucatán (CICY), Yucatán P. C. 97205, Mexico</footnote>

<footnote n="4">⁴ Consejo Nacional de Ciencia y Tecnología (CONACYT), Mexico City P. C. 03940, Mexico</footnote>

<footnote n="5">⁵ Department of Automatic Control, CINVESTAV-IPN, Av. Instituto Politécnico Nacional 2508, Mexico City 07360, Mexico; belem.saldivar@cinvestav.mx</footnote>

<footnote>* Correspondence: damien.guilbert@univ-lorraine.fr (D.G.); victor.ramirez@cicy.mx (V.R.)</footnote>
Today, different manufacturers offer many options and types of FCs. They can be classified according to their specific characteristics, such as the type of fuel used, the reaction temperature, and the electrochemical material used [5]. In addition, the main FCs reported in the literature are proton exchange membrane (PEMFCs), direct methanol, solid oxide, molten carbonate, phosphoric acid, alkaline, and microbial FCs [6,7]. In the field of both stationary and mobile applications, PEMFCs are among the most widely used and promising. This is due to their main features, which are quick and silent start-up operation, robustness, and a relatively low operating temperature range, which is, in general, between 60 and 80 °C [8,9].

Despite the significant technological advancements in PEMFC in recent years, issues such as their cost, size, weight, and complexity of peripheral devices are still of interest for research because the PEMFC system is intricate, involving several physical phenomena, including thermal, electrochemical, and electrical aspects. The PEMFC stack needs several auxiliary components for its correct operation, such as a humidification system, a cooling system, an air management system, and a hydrogen supply system. To enhance PEMFC performance, it is necessary to study, develop, and optimize every auxiliary component. This will allow the optimization of both the PEMFC stack and the powerful nonlinear interactions that exist among its components [10]. However, the acquisition of PEMFC system components and the considerable amount of labor required to assemble the usual complex PEMFC systems result in very high costs [11]. Therefore, due to the high costs involved, many researchers have performed studies based on numerical simulations using computational fluid dynamics (CFD), which is a branch of fluid mechanics that uses numerical analysis and data structures to analyze and solve problems involving fluid flows in PEMFCs. However, the experimental hardware is ignored when using simulation (software) only, but this hardware is important in studying the behavior of the PEMFC in a complete system (microgrid). For this reason, hardware capable of accurately imitating the behavior of a PEMFC can allow experimentation without the use of a real PEMFC, at least for the first experimental stages. These hardware systems are called emulators or real-time simulators [12].

A real PEMFC can be replaced with an emulator, allowing for the study and configuration of the remaining PEMFC components (auxiliaries, power electronics interfaces, loads, etc.) [13]. The final stage involves replacing the emulator with the real PEMFC once the system has been thoroughly examined and confirmed. In this way, all risks of damage to the system are reduced, and money, time, and space are saved [14].

Mathematical models are essential for the development of PEMFC emulators [15]. However, despite the wide variety of models developed for PEMFCs [16–25], those that are most suited to the purpose of emulator development are the equivalent electronic circuit models (ECMs) because these models are fit to describe the electrical behavior of a PEMFC or how the PEMFC interacts with the associated electrical systems and power conditioning circuits such as power electronics [26]. In addition, ECMs aid with electronic interface design and control and reliability test analysis [27,28]. Two types of ECMs have been reported: dynamic and passive models [26]. Chemical and thermodynamic processes must both be considered in a dynamic model. Consequently, it is possible to construct an optimal system in terms of efficiency and cost by understanding the special properties of the PEMFC [29,30]. Furthermore, power converter performance, transient response, and efficiency can be improved using a dynamic model, allowing for the creation of control systems that are appropriate for the load demand [31]. Passive models can be used to predict the performance and degradation of a PEMFC while it is in standby mode. This mode is appropriate for uninterruptible power systems when dependability is crucial and the PEMFC is idle for the majority of the time [26].

With the development of ECMs for PEMFCs, the construction of emulators has become viable, because it depends on the construction and implementation of an electronic circuit into the system. Although some authors have already developed and implemented emulators, this field is still developing due to the few reported studies in the literature. The re-
ported emulators can be classified as pseudo emulators [10,32] and electronic emulators, the latter of which are divided into electrical [11,12,14,15,33–66] and heat emulators [11,36, 55,67].

Particularly for electrical emulators, different investigations have been carried out for their design, construction, and validation. In [59–61,63,66], electrical emulators have been developed for 10 W, 1.2 kW, 3 kW, and 5 kW PEMFC systems. Additionally, the reported controllers for electric emulators have been realized on hardware platforms based on DC–DC buck converters, the control strategies of which are mainly based on PI controllers [49] and fuzzy control [54], among others. This kind of system is typically implemented on digital platforms using digital signal processors (DSPs) or dSPACE control boards [10,14,57,62] and field programmable gate arrays (FPGAs) systems [61]. For the validation of the emulators, in addition to performing a comparison of data taken from a real PEMFC, the hardware-in-a-loop (HIL) system has been successfully used [37,40–42,44,46–48,56]. As a result, due to the few reported studies and the importance of PEMFC emulators for the investigation and optimization of PEMFC systems, this review aims to provide a guide for the development of future emulators for PEMFCs.

This paper is composed of six sections. After pointing out the motivations and current state of the art regarding this review in the Introduction, in Section 2, a summary of the basic PEMFC characteristics is presented. After that, the different types of emulators, controllers, and validation methods are explained in Section 3. In Section 4, mathematical models of PEMFC emulators are described. Because the design of electronic circuits is important for the construction and implementation of an electronic emulator PEMFC, examples of these designs are presented in Section 5. Finally, in Section 6, a discussion is given.

## 2. The Basic Operation of PEMFCs

A lot of research has been conducted to make PEMFCs that are highly reliable and efficient for use in various applications such as portable power source devices and stationary or mobile applications. Owiwang to the development of computational fluid mechanics and working memory resources in computers capable of simultaneously solving many equations, recent advances have been made, particularly in materials and current density, which will eventually increase power density, device efficiency, and reliability [68]. In addition, compared with heat engines and when used in modular power generation, PEMFCs are more efficient [69].

A PEMFC is formed by a proton exchange membrane sandwiched between two electrodes (anode and cathode). Due to its unique characteristic, the membrane only permits protons to travel through while blocking electrons, especially perfluoro sulfonic acid membrane materials represented by Nafion and other sulfonated polymers [70]. In particular, Nafion contains a hydrophobic polytetrafluoroethylene (PTFE) backbone and a hydrophilic sulfonic acid group as the end group of the side chain. The membrane phase separates only after hydration, forming channels for proton conduction [71]. As hydrogen gas travels over the anode, it splits into hydrogen protons and electrons with the aid of a catalyst. Electricity is produced by the flow of protons via the proton exchange membrane to the cathode and the flow of electrons through an external circuit. Water is created when oxygen and hydrogen protons and electrons pass through the cathode, as shown in Figure 1. The reactions in a PEMFC are given by [72]:

$$ 
\begin{aligned}
2H_2 & \rightarrow 4H^+ + 4e^- \quad (\text{anode reaction}) \\
O_2 + 4H^+ + 4e^- & \rightarrow 2H_2O \quad (\text{cathode reaction}) \\
2H_2 + O_2 & \rightarrow 2H_2O \quad (\text{overall reaction})
\end{aligned} 
\quad (1) $$

The typical characteristics of PEMFCs are usually described by the polarization curve, which is a function depending on the voltage and current of the cell. Due to PEMFC electrical impedance, ineffective transport of the reactant gas, and slow reaction rate, the voltage decreases as more current is extracted from it. Low-load operation is desirable because a lower voltage indicates a lower-efficiency PEMFC. However, FC size and weight
are increased as a result. Furthermore, mobile applications that require frequent load changes cannot operate at low loads all the time. The polarization curve varies with the operating conditions, including pressure, temperature, partial pressure of reactants, and membrane humidity [69].

<figure><img src="image_4.png" /><figcaption>**Figure 1.** Scheme of a basic PEMFC.</figcaption></figure>

As shown in Figure 1, to form a complete FC system, the PEMFC stack needs several ancillary parts in addition to the four main flow subsystems: an anode hydrogen supply system, a cathode air supply system, a humidifier, and a cooler that maintains the temperature and humidity level of the PEMFC. To avoid constant heating and ensure fast system transient response, safe shutdown, system robustness, and the ability to adapt to power changes, the main parameters that need to be regulated are reagent flow rate, total pressure, partial pressure of the reagent, membrane, temperature, and humidity. The key control mechanisms are the water pump for temperature regulation, the humidifier for humidity management, the hydrogen flow and pressure-regulating valve, and the compressor motor for pressure regulation and airflow. It is worth mentioning that changes in one parameter influence the others. For instance, an increase in airflow velocity can raise the air pressure, but it can also change the amount of heat and steam that enters and leaves the stack, influencing the temperature and humidity of the stack and the membrane [57,69]. Thus, this is why PEMFC systems are complex.

### 3. Types of PEMFC Emulators

Emulators are frequently employed in renewable energy systems because these electrical-power-generating systems are expensive, typically have security requirements, use chemical reagents, and exhibit unpredictable behavior [38]. Emulators for different types of fuel cells have already been reported in the literature [73–75]. However, as mentioned in the Introduction, this review is focused on PEMFC emulators.

PEMFC emulators are designed to reproduce certain characteristics of this type of FC. Thus, experiments and studies of these characteristics have been carried out mimicking real performance without the need for a PEMFC [5,10]. Currently, an efficiency of 97% has been achieved for PEMFC emulators [5]. The initial step in simulating PEMFC systems is establishing the emulator’s implementation characteristics while taking into account the PEMFC application and objective. It is recommended to use digital processing systems as they allow quick and easy modifications of the algorithm because the emulation system must be frequently modified [38]. Additionally, for emulator lifetime purposes, several
factors must be taken into account, such as the build quality, the type of components, the way it is used and maintained, the deployment environment, and the frequency of use. Additionally, PEMFC emulators are subject to wear and tear and possible failure due to continued use or the passage of time. In this section, different types of PEMFC emulators are briefly described.

### 3.1. Pseudo Emulators

PEMFC pseudo emulators are designed to replace computer models of PEMFCs with a real small PEMFC. With the use of scaling rules, the emulator is capable of imitating the behavior of a full-size PEMFC using this method. Pseudo emulators have the benefit of scaling currents and voltages in response to the emulator's actual load, which enables the testing of systems of any size [32]. However, this type of emulator requires suitable PEMFC auxiliary components [10].

### 3.2. Electrical Emulators

By programming a DC power source, a PEMFC electrical emulator can be quickly and easily created. Nevertheless, these DC power sources frequently have current and voltage ranges that are not the same as those needed to replicate a particular PEMFC model [63]. Therefore, a DC transformer based on a noninverting DC-DC buck-boost converter (generally buck converter) is necessary between the load and the DC power source to modify the voltage or current of the emulated PEMFC [34,63].

Considering this electronic emulator development, the use of control strategies and implementation methods is necessary for the reliable validation of PEMFC electrical emulators. For this reason, this subsection also presents some control techniques reported in the literature.

## Control Strategies and Implementation Methods for Electric Emulators

**Proportional-integral (PI) controllers.** PI controllers have been applied to control electric emulators based on buck converters. This controller creates a control signal for a pulse width modulator (PWM), which provides the gate signal pulses of a MOSFET switch by comparing the output voltage to the PEMFC's reference voltage [49].

**Fuzzy controller.** These models have been proven to be suitable for the implementation of PEMFC emulators and the development of the simulation of control strategies due to their simplicity, short processing time, and extensive knowledge of implementation techniques [54]. The fuzzy relational model, neural-network-based fuzzy model, T-S fuzzy model, and fuzzy basis function-based model are a few modeling methods based on fuzzy reasoning that have been presented in recent years [76]. In particular, for PEM emulators, given their computing limitations, a low-cost digital processing device is created to apply a fuzzy-based model in the emulator and meet the needs of real-time processing [54].

**Digital signal processor (DSP) and dSPACE.** In the application of PEMFC emulators, a linear power amplifier linked to a DC-DC converter or directly to a load uses the output voltage from dSPACE as a reference control. The dSPACE protoboard has a digital-to-analog converter to connect the reference cell voltage from the PEMFC model to the power amplifier, an integrated DSP where the PWM controller is loaded, an analog-to-digital converter to read the measurement inputs from the sensors, and a digital input/output port for sending and receiving PWM signals [10,14,57,62].

**Field-programmable gate array (FPGA).** After production, an FPGA-integrated circuit can be configured by a customer or designer. In PEM emulator applications, the digital controller that simulates the PEMFC stack being tested is part of the FPGA system. The controller receives the digitized output voltage from the digitized current stream. The power stage provides the interface between the charging device and the emulator system. The controller samples the FC current while simulating the voltage across the PEMFC. Additionally, using the controller, it is possible to establish the initial temperature of the PEMFC stack
being tested as well as the ambient temperature and determine the temperature of the stack using current samples [61].

### 3.3. Heat Emulators

Studies on heat emulators based on electronic circuits are very scarce [11,36,67]. Electric plate heaters were used to generate the necessary amount of heat, emulating this parameter from the PEMFC stack. In addition, a collection of copper blocks was used to imitate a PEMFC stack’s internal heat capacity. To do this, the right number of blocks was selected to correspond to the comparable heat capacity of an actual PEMFC [11]. In the studies reported so far, the PEMFC heat emulators have been controlled by a microcontroller [11,36,67].

### 3.4. HIL Method

A PEMFC emulator can be used for HIL applications to test and validate the design of PEMFC systems [44]. The HIL method is a simulation/emulation process that connects software models of other system components with the hardware being tested or a reduced version of it [32]. Thus, using the HIL method, PEMFC auxiliaries can be tested and improved in real time with a PEMFC emulator without the risk of the PEMFC stack being damaged and at a low operating cost [44].

Figure 2 presents a diagram that summarizes the different types of reported emulators and the main characteristics of their design and construction.

<figure><img src="image_5.png" /><figcaption>Figure 2. Diagram of the different types of emulators and the main characteristics of their design and construction.</figcaption></figure>

## 4. Mathematical Model

The mathematical models used for the development of simulations, electronic circuits, and emulators are presented in this section.

### 4.1. Voltage-Current Models

Generally, the polarization curve of a PEMFC is non-linear and expressed in terms of PEMFC power $P_{fc}$:

$$P_{fc} = V_{fc} \cdot I_{fc} \qquad (2)$$
where $V_{fc}$ is the voltage (V), and $I_{fc}$ is the electrical current (A) of the PEMFC. The voltage $V_{fc}$ is usually expressed in terms of Nernst's voltage $E_{th}$ and the voltage drops: activation $V_{act}$, ohmic $V_{ohm}$, and concentration $V_{con}$ [14,77]:

$$V_{fc} = E_{th} - V_{ohm} - V_{act} - V_{con}, \qquad (3)$$

**Nernst's voltage** $E_{th}$. The difference between the reactant products and the Gibbs free energy yields Nernst's voltage, also known as the open-circuit voltage, which is the highest power obtained by one cell and corresponds to the exchanged Gibbs free energy. The following equation can be used to describe it [78,79]:

$$E_{th} = E_0 + B_1 \cdot (T_0 - T) + B_2 \cdot T \cdot \ln \left[ \frac{P_{H_2} \cdot P_{O_2}^{1/2}}{P_{H_2O}} \right], \qquad (4)$$

where $T_0$ and $T$ are initial and cell temperatures (K), respectively; $P_{H_2}$, $P_{O_2}$, and $P_{H_2O}$ are hydrogen, oxygen, and water pressures (atm), respectively; $E_0$ is the reference voltage (V); $B_1$ and $B_2$ are positive constants [77]. It should be noted that the initial pressure values in a PEMFC can be measured using differential pressure sensors, which are usually placed in the inlet and outlet flow channels of the gases. Thus, the pressure difference can provide information about the flow and resistance in a cell.

**Ohmic voltage drop** $V_{ohm}$. The ohmic voltage drop results from the resistances of the polymer membrane to proton transfer and of the electrode and collector plate to electron transfer [77–79].

$$V_{ohm} = I_{fc} \cdot R_{ohm}, \qquad (5)$$

where $R_{ohm}$ is the internal electrical resistance (Ω). The membrane conductivity $\sigma_m$ can be used to express the ohmic resistance ($cm \cdot \Omega^{-1}$).

$$R_{ohm} = \frac{t_m}{\sigma_m}, \qquad (6)$$

where $t_m$ is the membrane thickness (cm). $\sigma_m$ can be expressed as a function of membrane water content $\lambda_m$ and $T$.

$$\sigma_m = (b_1 \cdot \lambda_m - b_2) \cdot \exp \left[ b_3 \cdot \left( \frac{1}{T_0} - \frac{1}{T} \right) \right], \qquad (7)$$

where $b_1$, $b_2$, and $b_3$ are constants and are usually assessed empirically.

**Activation voltage drop** $V_{act}$. The requirement to transfer electrons, and break and create chemical bonds in the anode and cathode causes the drop in activation voltage. Driving the chemical reaction that moves the electrons to and from the electrodes uses up some of the available energy. Both the anode and cathode of a PEMFC electrode experience activation voltage. Yet, compared with the reaction of oxygen at the anode, the hydrogen oxidation process occurs quickly [77]. $V_{act}$ can be expressed as (8) [57,72].

$$V_{act} = \left( \frac{R \cdot T}{\alpha \cdot z \cdot F} \right) \cdot \ln \left( \frac{I_{fc}}{I_0} \right) = b_4 + T \cdot [b_5 + b_6 \cdot \ln(I_{fc})], \qquad (8)$$

where $R$ is the universal gas constant ($J \cdot mol^{-1} \cdot K^{-1}$), $\alpha$ is the electron transfer coefficient, $z$ is the number of participating electrons, $F$ is Faraday's constant ($C \cdot mol^{-1}$), and $I_0$ is the exchange current (A). $b_4$, $b_5$, and $b_6$ are the constants in the Tafel equation.

**Concentration voltage drop** $V_{con}$. The concentration gradients created by mass diffusions from the flow channels to the reaction sites are represented by the concentration voltage drop (catalyst area). High current densities, lethargic transport of reactants and products to and from the reaction sites, and a water film coating the catalyst surfaces on
the anode and cathode are the causes of this voltage drop [77]. $V_{con}$ can be expressed by (9) [11,72].

$$V_{con} = \left( \frac{R \cdot T}{z \cdot F} \right) \cdot \ln \left( 1 - \frac{I_{fc}}{I_{max}} \right), \qquad (9)$$

where $I_{max}$ is the maximum operating current of a PEMFC (A) with a restriction established by concentration losses.

## 4.2. Voltage-Current ECMs

Researchers conducted an electrochemical analysis to develop an ECM for a PEMFC, which is applicable for modeling power generation and its converters. Model parameters were determined using experimental polarization curves and electrochemical impedance spectroscopy, and simulation results were validated using these methods [28].

**Nernst's voltage.** This voltage can be emulated by a DC voltage source $V_{ini}$ [80]. Thus, it is possible to establish the following equation:

$$E_{th} = V_{ini}. \qquad (10)$$

**Ohmic voltage drop.** An equivalent resistance connected in series to the model PEMFC terminal can simulate this voltage. FC manufacturers provide the corresponding internal resistance value. However, it must be remembered that variations in operating temperature can cause changes in the resistance of the PEMFC [14]. Notice that $V_{ohm}$ can be represented by (5) as well.

**Activation and concentration voltage drops.** These voltages are responsible for the complex dynamics of the circuit. The double-layer charge effect serves as the primary regulator of PEMFC dynamics. A capacitor can be used to represent the charge layer that corresponds to the electrolyte/electrode contact because it functions as storage for electrical charges. Each change in voltage necessitates a charging time (in the event of an increase in voltage) or a leakage period (in the event of a voltage decrease). The ohmic drop, whose change can be thought of as instantaneous, is unaffected by this time delay but the activation and concentration voltages are affected. Thus, modeling of the activation and concentration voltage drops as first-order delay dynamics is possible.

$$\frac{d(V_{act} + V_{con})}{dt} = \frac{1}{C} \cdot I_{fc} - \frac{1}{\tau} \cdot (V_{act} + V_{con}), \qquad (11)$$

with a time constant $\tau = C \cdot R_a$. Given that the equivalent resistance $R_a$ depends on the activation and concentration voltages as well as the load current, the time constant $\tau$ controls the dynamics changes depending on the load conditions:

$$\tau = C \cdot R_a = C \cdot \left( \frac{V_{act} + V_{con}}{I_{fc}} \right). \qquad (12)$$

So, by using (5), (10), and (11), PEMFC voltage can be expressed as:

$$V_{fc} = V_{ini} - I_{fc} \cdot R_{ohm} - (V_{act} + V_{con}). \qquad (13)$$

## 4.3. Heat and Temperature Models

The thermal dynamic response of a FC $Q_{fc}$ is due to each cell layer's capacity to conduct heat. The following description fits this dynamic for each thermal control volume [57]:

$$\frac{dQ_{fc}}{dt} = \frac{dQ_c}{dt} - \frac{dQ_e}{dt} - \frac{dQ_{loss}}{dt}. \qquad (14)$$
where the chemical reaction $Q_c$, the electrical output power $Q_e$, and the heat loss $Q_{loss}$ are responsible for releasing the available power and can be estimated by (15) to (17), respectively [57].

$$ \frac{dQ_c}{dt} = n_{H_2} \cdot (\Delta G_0 - R \cdot T \cdot \ln(P_{H_2} \cdot \sqrt{P_{O_2}})), \qquad (15) $$

$$ \frac{dQ_e}{dt} = V_{fc} \cdot I_{fc}, \qquad (16) $$

$$ \frac{dQ_{loss}}{dt} = h \cdot N \cdot A_{cell} \cdot (T - T_r), \qquad (17) $$

where $n_{H_2}$ is the number of moles of hydrogen (mol), $\Delta G_0$ is the Gibbs free energy (J·mol⁻¹), $h$ is the coefficient of convective heat transfer (W·cm⁻²·K⁻¹), $N$ is the number of cells in the stack, and $A_{cell}$ is the cell area (cm²).

Finally, a PEMFC works at a constant temperature when $Q_{fc} = 0$; however, the temperature may increase or decrease during transitions according to (18) [81].

$$ M \cdot C_s \cdot \frac{dT}{dt} = \frac{dQ_{fc}}{dt}, \qquad (18) $$

where $M$ is PEMFC mass (kg), and $C_s$ is the equivalent average specific heat coefficient (J·kg⁻¹·K⁻¹).

### 4.4. Heat and Temperature ECMs

The heat emulator circuit is based on a controllable voltage source $V_H$ and a resistive heater $R_H$. The heat generated is a direct function of $V_H$ [11].

$$ Q_{fc} = \frac{V_H^2}{R_H}. \qquad (19) $$

In other words, $V_H$ directly controls the magnitude of the emulated waste heat.

### 4.5. Electrochemical Models

In this subsection, the equations that complement the emulation of a PEMFC system are presented. The electric power of the stack $P_S$ (W) is obtained via:

$$ P_S = N \cdot P_{fc}. \qquad (20) $$

The equations for calculating mass fractions are provided in this subsection due to their importance in PEMFC simulations. The equations used to calculate the mass fractions provided in this review are ordinary differential equations or 0D, which are validated by measuring the inputs and outputs of the masses in a PEMFC.

The hydrogen mass flow rate of oxygen $m_{H_2}$ (kg·s⁻¹) can be obtained using (21) [81].

$$ m_{H_2} = \left( \frac{M_{H_2}}{2 \cdot F} \right) \cdot \left( \frac{P_S}{V_{fc}} \right), \qquad (21) $$

where $M_{H_2}$ is the molar mass of hydrogen (Kg·mol⁻¹). Additionally, the hydrogen excess ratio $\mu_{H_2}$ is calculated as a function of the hydrogen mass flow rate entering anodes $m_{H_2,in}$ and $m_{H_2}$ [57].

$$ \mu_{H_2} = \frac{m_{H_2,in}}{m_{H_2}}, \qquad (22) $$
The mass flow rate entering the anode channel $m_{H_2,in}$ is calculated using the hydrogen mass flow rate reacting in the PEMFC stack and the purge mass flow rate $m_{H_2,purge}$. Therefore, $m_{H_2,in}$ can be defined by [57]:

$$m_{H_2,in} = m_{H_2} - m_{H_2,purge}. \qquad (23)$$

Moreover, the air mass flow rate $m_{O_2}$ ($kg \cdot s^{-1}$) is obtained using (24) [81].

$$m_{O_2} = \left( \frac{M_{O_2}}{4 \cdot F} \right) \cdot \left( \frac{P_S}{V_{fc}} \right). \qquad (24)$$

where $M_{O_2}$ is the molar mass of oxygen ($Kg \cdot mol^{-1}$). Additionally, the oxygen excess ratio $\mu_{O_2}$ is calculated as a function of the oxygen mass flow rate entering cathodes $m_{O_2,in}$ and $m_{O_2}$ [57].

$$\mu_{O_2} = \frac{m_{O_2,in}}{m_{O_2}}, \qquad (25)$$

$m_{O_2,in}$ is calculated using the mass flow rate of dry air $m_{a,in}$ in the cathode inlet and oxygen mass fraction $m_{O_2,fra}$,

$$m_{O_2,in} = m_{a,in} \cdot m_{O_2,fra}. \qquad (26)$$

Additionally, during the operation of the stack, the rate of water production $m_{H_2O}$ ($kg \cdot s^{-1}$) is calculated using (27) [81]

$$m_{H_2O} = \left( \frac{M_{H_2O}}{2 \cdot F} \right) \cdot \left( \frac{P_S}{V_{fc}} \right). \qquad (27)$$

where $M_{H_2O}$ is the water molar mass ($Kg \cdot mol^{-1}$).

Finally, the system PEMFC efficiency is obtained by (28) [81]

$$\eta = \frac{P_S}{m_{H_2O} \cdot LHV_{H_2}}, \qquad (28)$$

where $LHV_{H_2}$ is the lower heating power for hydrogen ($120 J \cdot kg^{-1}$). This parameter is a measure of the available thermal energy produced by the combustion of hydrogen in a PEMFC. This parameter is calculated as the subtraction of the heat of the vaporization of water from the higher calorific value.

## 5. Electronic Circuit Designs for Electronic Emulators

Because the design of electronic circuits is important for the construction and implementation of electronic emulators, some electronic circuit designs are presented in this section.

The first ECM for PEMFC was proposed by Larminie et al. [82]. In this model, each electrode is represented by a parallel-connected capacitor, resistance, and voltage. The circuits reported in the literature are mainly focused on describing the voltage behavior of PEMFCs. For the design of electronic circuits, some authors have employed mathematical models [4,26–28,30,72,83–100], while others have chosen to describe the polarization curve taking into account the behavior of the electronic components [29,31,101].

### 5.1. Design of Electronic Circuits for Electrical Emulators

Examples of electronic circuit designs for electrical emulators are presented in this subsection for both equation-based designs and electronic components.
5.1.1. Design of electronic circuit equation based electronic circuits

To design an electronic circuit based on voltage–current equations, Nernst’s voltage $V_{ini}$ and the voltage drops must be considered (see (3) and (13)), as depicted in Figure 3.

<figure><img src="image_6.png" /><figcaption>**Figure 3.** Schematic design of an electronic circuit using voltage–current equations.</figcaption></figure>

An example of an electronic circuit is presented in Figure 4. In this case, the authors considered the nonlinear behavior of voltage drops $V_{act}$ and $V_{con}$, which were modeled by a capacitor C (double-layer capacitance) and two resistances $R_1$ and $R_2$. One resistance $R_{ohm}$ was used to model the voltage drop $V_{ohm}$. Additionally, in this example, the relationship of the PEMFC voltage with temperature and current is considered [35].

<figure><img src="image_7.png" /><figcaption>**Figure 4.** Example 1 of electronic circuit design for an electrical emulator [35].</figcaption></figure>

Another design example of an electronic circuit is illustrated in Figure 5. This circuit consists of a PEMFC membrane resistance $R_{ohm}$, a parallel combination in series of one capacitor, and an impedance of a Faradic reaction formed of a charge transfer resistance $R_{ct}$ and a specific electrochemical element of diffusion, which is also called a Warburg element (this element is estimated using a series combination of two parallel RCs) [83].

<figure><img src="image_8.png" /><figcaption>**Figure 5.** Example 2 of electronic circuit design for an electrical emulator [83].</figcaption></figure>
One more example of electronic circuit design based on voltage-current equations is presented in [84]. The design includes the effects in both the anode (one capacitance $C_a$ and two resistances $R_{act,a}$ and $R_{con,a}$) and cathode (one capacitance $C_c$ and two resistances $R_{act,c}$ and $R_{con,c}$) sides, as shown in Figure 6.

<figure><img src="image_9.png" /><figcaption>Figure 6. Example 3 of electronic circuit design for an electrical emulator [84].</figcaption></figure>

However, as discussed in [84], compared with the cathode side, the anode side is barely affected by the activation and concentration voltage drops. To imitate this, a simplified ECM can be used, as shown in Figure 7.

<figure><img src="image_10.png" /><figcaption>Figure 7. Example 4 of electronic circuit design for an electrical emulator [84].</figcaption></figure>

### 5.1.2. Design of a Components-Based Electronic Circuit

In this example of electronic circuit design, the authors in [29,31] analyzed the behavior of the PEMFC polarization curve. Thus, they proposed a simplified electrical equivalent circuit built by a resistance $R_{fc}$ in series with a DC voltage source, as shown in Figure 8.

<figure><img src="image_11.png" /><figcaption>Figure 8. Example 5 of electronic circuit design for an electrical emulator [29,31].</figcaption></figure>

### 5.2. Design of Electronic Circuits for Heat Emulators

In the studies reported so far, a controllable voltage source $V_H$ and a resistive heater $R_H$ form the heat electronic circuit. Additionally, the heat produced using this circuit is due to $V_H$ (see (19)). The block of materials $Rt_{Body}$ that must take into account the internal thermodynamic characteristics is positioned between $R_H$ and the active cooling system,
as shown in Figure 9. The internal heat capacitances and thermal conductivity of stack membranes, gas diffusion layers, bipolar plates, and any other heat transfer physics that exist between the principal stack materials and the flowing coolant are some examples of these internal thermodynamic properties [11].

<figure><img src="image_12.png" /><figcaption>**Figure 9.** Example of electronic circuit design for a heat emulator [11].</figcaption></figure>

**6. Discussion**

Figure 10 shows a comparison of the number of studies reported for models and designs of electronic circuits and the number of studies for validated PEMFC emulators per year. Before 2010, there was the highest number of papers on the development of electronic circuits that imitate the behavior of a PEMFC, while for emulators validated for PEMFC systems, the highest number of reported studies was between the years 2008 and 2020. This demonstrates the maturity that this technology has achieved in recent years.

<figure><img src="image_13.png" /><figcaption>**Figure 10.** Evolution of the number of studies on electronic circuit designs and validated emulators per year.</figcaption></figure>

Figure 11 illustrates a summary of this work. For electronic emulators (electrical and heat), controllers and mathematical models (simulation) need to be connected to them to obtain reliable measurements of a PEMFC system. Forr pseudo emulators, adequate scaling is needed for a small PEMFC to collect data from a reduced PEMFC system; later, these data are scaled to a real PEMFC system. However, in addition to the different parameters proposed in this review, it is necessary to take into consideration the different cooling methods used for the development of emulators and complex simulations [102,103], which are usually air cooling for PEMFCs less than 5 kW and coolant cooling for PEMFCs with power greater than 5 kW. Therefore, the temperature parameter of the PEMFC and the end purpose of the emulator are important factors to consider when developing one. Figure 11 shows the methods used to
validate PEMFC emulators in general (i.e., through a comparison with data from a real PEMFC system and the HIL method).

<figure><img src="image_14.png" /><figcaption>**Figure 11.** Diagram of the interactions of the different types of emulators with their components for proper operation.</figcaption></figure>

## 7. Conclusions

PEMFC emulators are a valuable tool for developing PEMFC systems. This is because emulators reduce experimental cost, labor time, and installation space in the early stages of research, plus there is no risk of damaging the PEMFC system. Additionally, the PEMFC emulators presented in this paper can be widely used in various microgrid and hybrid energy storage simulators, where a fuel cell is an object (black box) in which input and output current parameters are read without considering the internal processes of the PEMFC.

This review presented a study of the different types of emulators for PEMFC (pseudo and electronic emulators). Electronic emulators have been developed in previous studies (development of ECMs); for their proper functioning, different types of controllers have been used. For pseudo emulators, an adequate reduction of the PEMFC system is necessary, which can be expensive. For the validation of PEMFC emulators, the data comparison method and the HIL method are generally used.

Different mathematical models were also presented for PEMFC systems. These models represent the simulated part of the system and, together with a PEMFC emulator, it is possible to obtain complete and reliable measurements of the entire system. Additionally, electrical circuit designs were presented to show the electrical components needed to build an electrical emulator (resistors, capacitors, and voltage sources). Therefore, this review supports the development of new PEMFC emulators.
**Author Contributions:** Conceptualization, Á.H.-G.; methodology, Á.H.-G., D.L.-C., and P.R.M.-R.; software, Á.H.-G. and V.R.; validation, D.L.-C., P.R.M.-R., and D.G.; formal analysis, Á.H.-G.; investigation, Á.H.-G. and B.S.; resources, D.G.; data curation, Á.H.-G., P.R.M.-R., and V.R.; writing—original draft preparation, Á.H.-G.; writing—review and editing, D.L.-C., P.R.M.-R., D.G., V.R., and B.S.; visualization, D.G. and V.R.; supervision, D.L.-C. and P.R.M.-R.; project administration, Á.H.-G.; funding acquisition, D.G. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research received no external funding.

**Institutional Review Board Statement:** Not applicable.

**Informed Consent Statement:** Not applicable.

**Data Availability Statement:** Not applicable.

**Acknowledgments:** This study was supported by Consejo Nacional de Ciencia y Tecnología (CONACYT) Mexico.

**Conflicts of Interest:** The authors declare no conflicts of interest.

**Abbreviations**

The following abbreviations are used in this manuscript:

<table><tr><td>DSP</td><td>Digital Signal Processor</td></tr><tr><td>ECM</td><td>Electronic Circuit Model</td></tr><tr><td>FC</td><td>Fuel Cell</td></tr><tr><td>FPGA</td><td>Field-Programmable Gate Array</td></tr><tr><td>HIL</td><td>Hardware-in-a-Loop</td></tr><tr><td>MDPI</td><td>Multidisciplinary Digital Publishing Institute</td></tr><tr><td>PEMFC</td><td>Proton Exchange Membrane Fuel Cell</td></tr><tr><td>PI</td><td>Proportional-Integral</td></tr><tr><td>PWM</td><td>Pulse Width Modulator</td></tr></table>

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