Acausal Fuel Cell Simulation Model for System Integration Analysis in Early Design Phases

Abstract

Hydrogen technologies have the potential to reduce aviation’s CO₂ emissions but come with many challenges. This paper introduces a scalable hydrogen fuel cell model tailored for system integration analysis in early aircraft design phases. The model focuses on Proton Exchange Membrane Fuel Cells (PEMFCs) and is based on thermodynamic equations and empirical data to simulate performance under different ambient and operating conditions; it also includes a simplified model of the Balance of Plant (BOP) systems and is implemented in OpenModelica. The model performance is validated through a comparison of the simulated polarization curves with real datasheet data. A case study highlights the peculiarities of this model by studying the sizing of the fuel cell stacks for a modified ATR 72 aircraft. The developed model effectively supports the early design exploration of the aircraft with a greater level of detail for system integration studies, essential to better explore the potential of aircraft featuring hydrogen-based power systems.

1. Introduction

Air traffic expansion is expected to double within the next two decades, increasing greenhouse gas emissions [1]. Aviation currently produces between 2 and 3.5 percent of these anthropogenic emissions [2], but, at the 41st assembly of the International Civil Aviation Organization (ICAO), 2050 was chosen as the time limit for reducing these emissions from aviation operations to zero. Among the strategies to reduce the production of CO₂ and other harmful substances, the IATA suggests using alternative fuels, such as hydrogen, burned in turbine engines or converted to electricity by fuel cells [3].

Hydrogen has been used in aviation since the 1700s, particularly in balloons and airships, and later explored for modern aircraft due to rising fuel costs [4,5,6]. The recent efforts focus on hydrogen combustion in turbines and fuel cells for hybrid-electric systems to reduce emissions. The use of hydrogen presents significant challenges for the design of new aircraft and systems. These include issues regarding safety, storage technology, and aircraft design changes. Companies such as ZeroAvia [7] and Airbus are advancing hydrogen aircraft technologies, intending to bring them to the market by the decade’s end. Overcoming these obstacles will require new design methodologies and models to better understand the behavior of these new technologies [4,5].

Fuel cells convert the chemical energy of hydrogen to electrical energy through layers of electrodes and an electrolyte. They are classified by their electrolytes and fuels, with Proton Exchange Membrane Fuel Cells (PEMFCs) and Solid Oxide Fuel Cells (SOFCs) being the prominent types. Single cells produce low voltage, so multiple cells are stacked together using bipolar plates or tubular cells [8]. The Balance of Plant (BoP) system supports fuel cell operation by managing cooling, water, reactant flow, and electrical conversion. These systems are essential for maintaining performance and enabling practical applications, such as transportation [9].

The fuel cell voltage varies with the operating conditions and decreases as the current increases. The maximum voltage, known as the open circuit voltage (OCV), occurs when no electrical load is applied. For low-temperature PEMFCs, the voltage initially drops rapidly, then decreases linearly and steeply at high currents. High-temperature fuel cells, like SOFCs, show a less significant initial drop. Voltage drops in fuel cells are due to several factors: activation losses, which are the energy required to start chemical reactions; fuel crossover and internal currents, which involve unwanted fuel leakage; ohmic losses, which are due to electron resistance and ion movement; and mass transport losses, which result from decreased reagent concentrations at high currents. These losses are shown in the polarization curve (Figure 1, an essential tool for fuel cell modeling) [10].

Numerical models of PEMFCs and SOFCs are crucial in designing innovative fuel cell systems due to their cost-effectiveness and ease of use compared to physical models and testing under real conditions [11]. The research on PEMFC models has evolved significantly. Early work by Bernardi and Verbrugge [12,13,14] focused on one-dimensional models for gas transportation and humidification requirements, later expanding to comprehensive models for cell performance and species transport. Springer et al. [15] introduced a one-dimensional steady-state model to study the diffusivities in the membrane, while van Bussel et al. [16] developed a two-dimensional dynamic model highlighting local humidity effects on current density.

The complexity of PEMFC models increased, incorporating three-dimensional models solving Navier–Stokes equations for deeper insights [17,18,19]. However, these models’ high computational demands led to the development of simplified models for practical applications in complex systems, such as those by Rowe and Li [20], focusing on temperature and water distribution effects, and Thanapalan et al. [21] and Kravos et al. [22], investigating PEMFC controllability. Commercial software like OpenModelica v1.23.1 and Simulink^® v9.1 facilitated the creation of robust and dynamic PEMFC models, such as FuelCellLib [23], and models by MathWorks^®, which offer varying levels of accuracy and flexibility, based on [24,25,26,27].

SOFC has also observed significant modeling advancements. Gebregergis et al. [28] developed an efficient lumped-parameter model for the real-time simulation and control of SOFCs using an equivalent resistor–capacitor circuit for dynamic responses. Costamagna et al. [29] presented a validated steady-state two-dimensional model showing a direct relationship between hydrogen and oxygen partial pressures and current density. Ho [30] created a dynamic three-dimensional model analyzing temperature, current density, activation overpotential, and hydrogen pressure transients. Bianchi et al. [31] validated a steady-state two-dimensional model based on electrochemical kinetics using experimental results from Electrochemical Impedance Spectroscopy.

Commercial CFD simulation or Model-Based Design (MBD) software often solves these models. Andersson et al. [32] created a one-dimensional SOFC model using Modelica, integrating electrolyte and fuel flow models. Lakshmi et al. [33] used a phenomenological approach to model a single SOFC with a transfer function to calculate the partial hydrogen, oxygen, and water pressures.

In recent years, fuel cell simulation models have been crucial in evaluating the performance of hydrogen-fuel-cell-powered systems in aviation, particularly with the development of simpler and faster models like the zero-dimensional electrochemical model by Vidović et al. [34], which includes a compressor and humidifier to assess the overall efficiency. The application of PEMFCs in aviation began with replacing individual components, such as APUs (Auxiliary Power Units), with studies by Pratt et al. [35] and Schröder et al. [36] showing potential performance benefits and optimization strategies. As PEMFC technology advances, it is being considered for primary electrical generation in all-electric aircraft. Kadyk et al. [37,38] and Hartmann et al. [39] emphasized comprehensive system analyses and innovative configurations, such as utilizing cryogenic hydrogen for cooling. Furthermore, Correa et al. [40] presented an additional dynamic model of a complete BoP and fuel cell, validated with aircraft flight tests, in which the authors integrated a fuel cell system. SOFCs offer flexibility with various fuels, improving the propulsion efficiency without major aircraft modifications. Studies by Gummalla et al. [41], Fernandes et al. [42], and Santarelli et al. [43] explored SOFCs as APUs and for combined electrical and thermal power generation. Hybrid systems, integrating SOFCs with Internal Combustion Engines (ICEs) and gas turbines, show promise for high-altitude and long-duration missions, as highlighted by Himansu et al. [44] and Collins and McLarty [45], who achieved improved power density with superconducting motors.

Numerous mathematical models have been documented in the literature, each capable of simulating fuel cell behavior at various levels of detail. The research works presented in this section are summarized in Appendix A. However, there is still room for a simplified dynamic model that can support the preliminary design of fuel cell power systems, with rapid usability in an open-source, acausal environment such as OpenModelica. Although previous work has developed simplified causal FC models in Simulink, their modification and integration with other systems are not as rapid as with an acausal model. This paper presents a scalable hydrogen fuel cell model for broader integration into aircraft systems using an equivalent electrical circuit approach compatible with Simulink’s Simscape and OpenModelica acausal environments. The model calculates the electrical and thermal power performance based on manufacturers’ datasheets without delving into the cell’s inherent electrochemical phenomena. Based on the work of Njoya Motapon [25], it refines the formulation of the partial pressures used to calculate the ideal voltage, adds mathematical models for heat transfer, and creates an acausal interface to support the flexible integration of fuel cell systems in aircraft within different simulation environments.

This first section introduced the use and modeling of fuel cells in sustainable aviation. Section 2 explains the mathematical model and its implementation in OpenModelica. Then, a case study is proposed in Section 3 to study the peculiarities of this model, and finally some considerations regarding the effectiveness and limitations of the model are presented in Section 4.

2. Methods

This section provides a detailed description of the mathematical model, beginning with a presentation of the equations that compose it. Next, the implementation of these relationships and a simplified BoP model in the OpenModelica development environment are discussed. Finally, the validation methodology of the fuel cell stack model is presented. For additional details regarding the model, please refer to [46].

Fuel cells generate electrical energy by converting the internal energy of reactants like hydrogen and oxygen. This process follows the two laws of thermodynamics, where (1) energy can never be created or destroyed but can be transformed from one form to another, and (2) entropy measures the possible states of a system related to reversible heat transfer and temperature [47].

Key thermodynamic potentials describe system behavior, i.e., internal energy (U), Gibbs free energy (G), and enthalpy (H). In fuel cells, the Gibbs free energy change ($\mathsf{\Delta }G$) determines the maximum electrical work ($W_{elec}$), with $W_{elec}=-\mathsf{\Delta }{G}_{rxn}$. The cell voltage (E) is provided by $E=-{\displaystyle \frac{\mathsf{\Delta }{G}_{rxn}}{zF}}$, where z is the number of electrons and F is Faraday’s constant. The Nernst equation, which accounts for temperature and concentration variations, provides the cell voltage:

$$E=E^0+{\displaystyle \frac{\mathsf{\Delta }{S}_{rxn}^0}{zF}}(T-T_0)+{\displaystyle \frac{RT}{zF}}ln{\displaystyle \frac{p_A{p}_B^b}{p_M^m{p}_N^n}}$$

(1)

For PEMFCs, typically operating around 100 °C, the parameters vary depending on whether the product water is gaseous or liquid.

Table 1. Reaction parameters for gaseous and liquid water products.

Parameter	Gaseous Water Product	Liquid Water Product
Reaction	$H_2+{\displaystyle \frac{1}{2}}{O}_2\to H_2{O}_{\left(g\right)}$	$H_2+{\displaystyle \frac{1}{2}}{O}_2\to H_2{O}_{\left(liq\right)}$
Reaction Enthalpy ($\mathsf{\Delta }{H}_{rxn}^0$)	−241.83 kJ/mol	−285.83 kJ/mol
Standard-State Entropy ($\mathsf{\Delta }{S}_{rxn}^0$)	−44.34 J/(mol K)	−163.23 J/(mol K)
Gibbs Free Energy ($\mathsf{\Delta }{G}_{rxn}^0$)	−228.61 kJ/mol	−237.16 kJ/mol
Standard Potential ($E^0$)	1.184 V	1.229 V

summarizes the key numerical results for both scenarios.

The potential of a fuel cell varies with pressure, temperature, and current load, combining these thermodynamic principles with models like the polarization curve to predict performance under different conditions.

The chosen fuel cell model is based on the equations developed in the work of Njoya Motapon [25]. These equations were modified to correctly describe the partial pressures, by Nehrir and Wang [48] equations, and the heat balance equation, by a simplified and detailed treatment of Spiegel [27]. Finally, they were implemented in an object-oriented modeling language to improve integration capabilities with other models employing acausal interfaces. Assumptions play a key role in defining the accuracy of the model, adapting it to the needs of the preliminary design stages, and reducing computational complexity. In particular, the model adopted here is zero-dimensional and simplifies the spatial complexity of fuel cell performance. Assumptions of this zero-dimensional model include ideal gas behavior, hydrogen and air supply, humidity and temperature maintained at optimal levels by dedicated systems, and negligible pressure losses. Voltage drops are attributed to reaction kinetics and ohmic losses, while mass transport effects at high current densities are neglected. The latter hypothesis is further explored in [37].

The model of a fuel cell stack is represented by an electrical equivalent circuit (Figure 2), comprising a controlled voltage source E, a resistor $R_{ohm}$, an ideal diode, and a current sensor. The voltage E is determined by

$$E=E_{oc}-NAln\left({\displaystyle \frac{i_{fc}}{i_0}}\right)·{\displaystyle \frac{1}{s{T}_d/3+1}}$$

(2)

Here, $E_{oc}$ is the open circuit voltage considering operating conditions, N is the number of series cells, A is the Tafel slope, $i_{fc}$ is the stack current, $i_0$ is the exchange current, and $T_d$ is the stack settling time.

The Tafel equation characterizes activation losses, accounting for nonlinear polarization at low currents. The model considers the dynamic Double-Layer Charge effect with a first-order transfer function, introducing a delay characterized by $T_d\approx 3\tau$, where $\tau$ is the time constant.

The total voltage across the fuel cell $V_{fc}$ is provided by

$$V_{fc}=E-R_{ohm}{i}_{fc}$$

(3)

The model allows input variations (stack temperature, pressures, composition, and flow rates), influencing parameters in E. It includes calculation blocks (A, B, and C) managing these inputs, with adjustments and extensions (Blocks D and Simplified D) to enhance thermal power calculation capabilities based on different approaches.

2.1. Block A: Rates of Utilization

Block A computes the utilization rates of hydrogen and oxygen in a fuel cell based on operational parameters and current supplied:

$$U_{f_{H_2}}={\displaystyle \frac{RTN{i}_{fc}}{zF{P}_{fuel}{V}_{fuel}x}},U_{f_{O_2}}={\displaystyle \frac{RTN{i}_{fc}}{2zF{P}_{air}{V}_{air}y}}$$

(4)

These rates denote the ratios of reacting moles to total moles entering the gas channels, which are fundamental for calculating subsequent partial pressures.

2.2. Block B: Open Circuit Voltage and Exchange Current

Block B calculates the open circuit voltage $E_{oc}$ and exchange current $i_0$. In the original model, it used the following formulations for partial pressures:

$$P_{H_2}=(1-U_{f_{H_2}})x{P}_{fuel},P_{O_2}=(1-U_{f_{O_2}})y{P}_{air},P_{H_{2}O}=(w+2y{U}_{f_{O_2}})P_{air}$$

(5)

However, these equations, sourced from [25], reflect partial pressures after the chemical reaction. The equations are then modified following the formulation expressed in [48]. The revised equations for partial pressures are

$$P_{H_2}=x{P}_{fuel},P_{O_2}=y{P}_{air},P_{H_{2}O}=w{P}_{air}$$

(6)

These revised equations represent the partial pressures at the fuel cell inlet, which are crucial for subsequent calculations in the model.

The Nernst voltage $E_n$ depends on the operational temperature T

$$E_n=\left\{\begin{array}{cc}1.229\mathrm{V}+(T-298\mathrm{K}){\displaystyle \frac{-163.23\mathrm{J}/\mathrm{mol}}{2F}}+{\displaystyle \frac{RT}{2F}}ln\left(P_{H_2}{P}_{O_2}^{0.5}\right)\hfill & \mathrm{if}T<373\mathrm{K}\hfill \\ 1.184\mathrm{V}+(T-298\mathrm{K}){\displaystyle \frac{-44.34\mathrm{J}/\mathrm{mol}}{2F}}+{\displaystyle \frac{RT}{2F}}ln\left({\displaystyle \frac{P_{H_2}{P}_{O_2}^{0.5}}{P_{H_{2}O}}}\right)\hfill & \mathrm{if}T\ge 373\mathrm{K}\hfill \end{array}\right.$$

(7)

The final output equations are   

$$E_{oc}=K_c·E_n,i_0={\displaystyle \frac{zFk(P_{H_2}+P_{O_2})\mathsf{\Delta }v}{Rh}}exp\left({\displaystyle \frac{-\mathsf{\Delta }G}{RT}}\right)$$

(8)

where $K_c$ is the voltage constant, k is Boltzmann’s constant, $\mathsf{\Delta }v3$ is the activation barrier volume factor, h is Planck’s constant, and $\mathsf{\Delta }G$ is the activation energy barrier. The differences between the original and modified versions mainly consist of the formulation of partial pressures and their respective application in calculating Nernst voltage, ensuring an accurate representation of fuel cell operating conditions.

2.3. Block C: Tafel Slope

Block C calculates the Tafel slope A with only the operational temperature T as input:

$$A={\displaystyle \frac{RT}{z\alpha F}}$$

(9)

where $\alpha$ is the charge transfer coefficient, i.e., the percentage of supplied electrical energy devoted to altering the rate of an electrochemical reaction [10].

2.4. Block D: Thermal Model

The model assumes sufficient cooling to maintain a constant temperature inside the fuel cell, which is critical for various power levels, and to remove the heat generated. Different cooling methods could be employed, as analyzed by Zhang and Kandlikar [49] and Bargal et al. [50], but liquid cooling systems are predominantly used for high-power aviation fuel cell stacks. These systems manage heat generated within the fuel cell, arising from reversible heat (entropic heat), irreversible heat from electrochemical reactions, ohmic heat due to resistance in ion and electron flow, and minor contributions from heat due to water vapor condensation.

To model heat generation, consider two approaches from Spiegel [27]: simplified and detailed. The energy balance of a fuel cell stack is expressed as

$$\sum Q_{in}-\sum Q_{out}=W_{el}+Q_{dis}+Q_c$$

(10)

Here, $Q_{in}$ and $Q_{out}$ represent the enthalpies of reactants and products, $W_{el}$ is the electrical power generated, $Q_{dis}$ is the dissipated heat to the environment, and $Q_c$ is the heat removed through the cooling system. Most of the heat is managed by the cooling system, either through exhaust or convection.

A simplified model for heat generation ($Q_{gen}$) assumes

$${\displaystyle \frac{I_{fc}}{2F}}HN=Q_{gen}+I_{fc}{V}_{cell}N$$

(11)

This model forms the basis of the simplified Block D and depends on operational temperature. For temperatures below 100 °C, $H=HH{V}_{H_2}=286$ kJ/mol, representing the complete combustion of hydrogen-producing liquid water. For temperatures above 100 °C, $H=LH{V}_{H_2}=242$ kJ/mol, considering the energy released by the complete combustion of hydrogen, accounting for water vaporization.

A more accurate energy balance considers the enthalpies of substances entering and leaving the fuel cell system:

$$\sum {\left(h_i\right)}_{in}=W_{el}+\sum {\left(h_i\right)}_{out}+Q_{gen}$$

(12)

Inputs include fuel, oxidant, and water vapor enthalpies at the inlet. Outputs include the enthalpies of exhausted fluids, electrical power, and heat dissipated by convection, radiation, or the cooling system. For the calculation of enthalpies, reference can be made to [27].

Figure 3 illustrates the four blocks implemented in OpenModelica. A detailed discussion regarding the implementation and parameter acquisition for the model equations is found in subsequent sections.

2.5. Model Parameters

The mathematical model calculates deviations from nominal conditions using parameters from the manufacturer’s datasheet. Key parameters include $E_{oc}$, N, $T_d$, $i_0$, $R_{ohm}$, $\alpha$, $\mathsf{\Delta }G$, and $K_c$. These are derived through specific equations, often requiring intermediate calculations shown in

Table 2. Equations for key parameters in the fuel cell mathematical model.

Parameter	Equation
Exchange Current $i_0$	$i_0=exp\left({\displaystyle \frac{V_1-E_{oc}-R_{ohm}}{NA}}\right)$
Ohmic Resistance $R_{ohm}$	$R_{ohm}={\displaystyle \frac{V_{nom}+NAln\left(I_{nom}\right)-V_1}{1-I_{nom}}}$
Factor $NA$	$NA={\displaystyle \frac{(V_1-V_{min})(1-I_{nom})+(1-I_{max})(V_{nom}-V_1)}{ln\left(I_{max}\right)(1-I_{nom})-(1-I_{max})ln\left(I_{nom}\right)}}$
Tafel Slope $\alpha$	$\alpha ={\displaystyle \frac{NR{T}_{nom}}{zFNA}}$
Partial Pressure $P_{H_2\left(nom\right)}$	$P_{H_2\left(nom\right)}=x_{nom}{P}_{fuel\left(nom\right)}$
Partial Pressure $P_{O_2\left(nom\right)}$	$P_{O_2\left(nom\right)}=y_{nom}{P}_{air\left(nom\right)}$
Parameter $K_1$	$K_1={\displaystyle \frac{2Fk(P_{H_2\left(nom\right)}+P_{O_2\left(nom\right)})P_{std}\mathsf{\Delta }v}{Rh}}$
Activation Energy $\mathsf{\Delta }G$	$\mathsf{\Delta }G=-R{T}_{nom}ln\left({\displaystyle \frac{i_0}{K_1}}\right)$
Nernst Voltage $E_{n\left(nom\right)}$	$E_{n\left(nom\right)}=1.229+(T_{nom}-298){\displaystyle \frac{-163.23}{zF}}+{\displaystyle \frac{R{T}_{nom}}{zF}}ln\left(P_{H_2\left(nom\right)}{P}_{O_2\left(nom\right)}^{0.5}\right)$
Dimensionless Parameter $K_c$	$K_c={\displaystyle \frac{E_{oc}}{E_{n\left(nom\right)}}}$

.

Using the polarization curve, values such as $E_{oc}$, $V_1$, $V_{nom}$, $I_{nom}$, $V_{min}$, and $I_{max}$ are extracted. Additional data from the datasheet provide information on the number of cells (N), nominal efficiency (${\eta }_{nom}$), operating temperature ($T_{nom}$), air flow rate ($V_{air\left(nom\right)}$), supply pressures ($P_{fuel\left(nom\right)}$, $P_{air\left(nom\right)}$), response time ($T_d$), and hydrogen, oxygen, and water mole fraction ($x_{nom}$, $y_{nom}$, $w_{nom}$).

2.6. Model in OpenModelica

The fuel cell model is implemented in the Modelica language within the OpenModelica environment. The model is encapsulated in the block Fuel_Cell_Core (Figure 4), which includes inputs, outputs, parameters, and acausal interfaces essential for simulation.

The inputs to the fuel cell model are organized into four record elements representing the inlet and outlet conditions of the cathode and anode circuits. Each record structure includes variables for pressure, mass flow rate, density, temperature, specific enthalpy, and mass fraction of each component in the gas mixture. To avoid numerical errors, limiters are applied to these inputs, ensuring that they stay within reasonable ranges, with minimum values specified for temperature (200 K), pressure (1 Pa), mass flow rate (1 × ${10}^{-4}$ kg/s), density (1 × ${10}^{-4}$ kg/m³), specific enthalpy (1 × ${10}^{-4}$ J/kg), and current (1 × ${10}^{-3}$ A). Mass fractions of the component are converted into mole fractions.

The block outputs the mass flow rates of hydrogen (${\dot{m}}_{H_2}$), oxygen (${\dot{m}}_{O_2}$), and water (${\dot{m}}_{H_{2}O}$) produced at the cathode. These are calculated using the following equations:

$${\dot{m}}_{H_2}=-M_{H_2}N{\displaystyle \frac{I_{fc}}{2F}}{\dot{m}}_{O_2}=-M_{O_2}N{\displaystyle \frac{I_{fc}}{4F}}{\dot{m}}_{H_{2}O}=M_{H_{2}O}N{\displaystyle \frac{I_{fc}}{2F}}$$

(13)

where $M_{H_2}$, $M_{O_2}$, and $M_{H_{2}O}$ are the molar mass values of hydrogen, oxygen, and water.

The acausal interfaces in the model include electrical poles for the electrical connection and a heat transfer interface for temperature (K) and heat flow (W).

In Block B, partial pressures of hydrogen, oxygen, and water are computed in sub-blocks BlockB3, BlockB2, and BlockB1. The Nernst voltage is calculated in BlockB5 with different equations depending on the temperature and is converted to the open circuit voltage in BlockB6 by multiplying it by $K_c$. The exchange current is obtained in BlockB4.

Block C calculates the Tafel slope, which depends on temperature and the parameter $\alpha$. The activation drop is determined using the fuel cell stack current, exchange current, and Tafel slope. This loss is subtracted from the open circuit voltage, and the result is provided as input to the voltage source of the equivalent electrical circuit. A resistor represents ohmic losses, a diode ensures current flows in one direction, and sensors measure voltage and current.

Block D calculates the fuel cell temperature using voltage and current inputs. If a cooling system is present, it connects to this block. Heat is calculated and fed into a thermal circuit.

2.7. PEMFC Stack Block

The PEMFC Stack block has acausal electrical and thermal circuit interfaces. It converts causal input and output elements. The anode and cathode circuits are modeled using the OpenModelica fluid library. The anode is a mixture of hydrogen and nitrogen, while the cathode is a mixture of nitrogen, oxygen, and water. These mixtures are defined in Modelica packages FuelMixture and AirMixture.

2.8. Model of the Balance of Plant

The BoP model implemented in OpenModelica comprises several components: a hydrogen source, exhaust conditions on the anode side, and a circuit on the cathode side to maintain proper pressure and mass flow. The mission profile model controls power demand and atmospheric conditions. The cooling system is absent, assuming a constant operating temperature. The BoP focuses on the fuel cell stack parameters based on power requirements for aircraft applications. A database for low- and medium-power applications is created, and, for higher-power needs, a fictitious fuel cell can be created by scaling existing ones. The power of the fuel cell stack is defined as

$$P_{stack}=V_{cell}·N·i·A_{cell}$$

(14)

A power requirement can be achieved by increasing the number of cells in series N or the cell area $A_{cell}$, each method having its pros and cons [37].

Hydrogen is supplied by a source with a constant mass flow rate on the anode side, determined by Equation (16). The cathode side model is more detailed, using a block to define air inlet conditions based on altitude and flight speed, following the International Standard Atmosphere (ISA) model. Air is compressed and maintained at a constant flow rate determined by Equation (17). The pressure relief valve ensures correct fluid pressure in the cathode gas channel.

The ISA atmosphere block calculates atmospheric conditions based on altitude and speed, providing static pressure and temperature.

The mission profile block simulates different flight stages by receiving a table of key mission points, interpolating between them to define conditions over time. This block outputs power, velocity, and altitude, feeding into the atmospheric conditions block and an electrical input block that determines the required current for the fuel cell stack, defined as

$$i_{fc}={\displaystyle \frac{P}{V_{fc}}}$$

(15)

This block is then directly connected to the fuel cell and grounded.

2.9. Validation

To validate the fuel cell stack model and not the simplified BoP, both PEMFC datasheets are used since many manufacturers provide polarization curves and other detailed product information, and experimental results are used. A comprehensive database of information on commercially available fuel cells was created to extract polarization curve data and analyze trends in energy density as power increases [46].

The validation setup involves creating a Balance of Plant (BoP) around the fuel cell block to simulate and compare the polarization curves from the datasheet and the simulation. The BoP includes simplified oxygen and hydrogen circuits and an electrical circuit, while the thermal circuit is neglected. For the hydrogen circuit, the mass flow rate is set to

$${\dot{m}}_{\mathrm{fuel},\max }={\displaystyle \frac{{\lambda }_{{\mathrm{H}}_2}{V}_{\min }{I}_{\max }}{{\eta }_{\mathrm{nom}}\mathsf{\Delta }{h}_{\mathrm{HHV}}}}$$

(16)

where ${\lambda }_{{\mathrm{H}}_2}=1.1$ is the stoichiometric ratio and $\mathsf{\Delta }{h}_{\mathrm{HHV}}=141.8\mathrm{MJ}/\mathrm{kg}$ is the enthalpy of hydrogen formation. For the air circuit, the mass flow rate is

$${\dot{m}}_{\mathrm{air},\max }={\displaystyle \frac{N{I}_{\max }{\lambda }_{{\mathrm{O}}_2}{M}_{\mathrm{air}}}{4F{y}_{{\mathrm{O}}_2}}}$$

(17)

where ${\lambda }_{{\mathrm{O}}_2}=1.8$ is the stoichiometric ratio. The electrical circuit consists of a current source ramping up to the maximum current defined in the datasheet and a circuit ground for reference.

Three different commercial fuel cell stacks were used to validate the model, whose polarization curves were compared with the simulation results. Further comparison was conducted by comparing the simulation results with experimental data [51]. The results are summarized in

Table 3. Validation results of fuel cell models.

Fuel Cell Model	Maximum Relative Error	Maximum Error Current Range
NedStack P8 PS6 [25]	3.2%	High currents
Horizon H5000 [52]	2.4%	Low currents
Horizon H1000XP [52]	3.8%	Low currents
NEXA 1200 [51]	5.5%	Low currents

.

The model approximates fuel cell polarization curves with low error, as can also be observed in Appendix B, validating its use for studying the integration of PEMFCs into system architectures. Accuracy depends on careful selection of input parameters.

3. Results: Case Study for Regional Aircraft Fuel Cell Scaling

The case study examines a modified ATR 72-based aircraft with cryogenic propulsion and a fuel cell stack powering two electric motors [39]. The 70-passenger aircraft, with a maximum takeoff weight of 23 tons, flies the Milan–Barcelona route (725 km). The mission profile, depicted in Figure 5, involves a low-power taxi phase followed by a short takeoff utilizing a maximum power output of 4.1 MW. Subsequently, a gradual reduction in throttle is implemented during the aircraft’s climb to an altitude of 6 km. Upon the completion of the climb, the power output stabilizes at 2.5 MW for the duration of the cruise, during which the aircraft maintains a velocity of 520 km/h for slightly over one hour. Following this, the power is incrementally decreased during the descent and landing phases. Notably, the currently available fuel cell stacks with applications in mobility are insufficient to meet these power demands as they are approximately an order of magnitude less powerful. Consequently, it is necessary to scale up an actual fuel cell stack to satisfy the power requirements by combining multiple fuel cells in series.

The fuel cell developed by ZeroAvia has been chosen as the primary device for this study, representing the cutting edge of fuel cell technology in aerospace applications [53]. This fuel cell has a maximum power output of 400 kW and serves as a critical model for testing the scalability of fuel cell technology in aviation. By using this device, the research aims to explore the challenges of scaling fuel cells to meet higher power demands. To accurately evaluate the performance of this fuel cell stack, the derivation of its polarization curve is essential. This study will use this curve to analyze various scaling factors and assess the fuel cell’s efficiency under different conditions. For a comprehensive discussion on the polarization curve and an additional case study, please refer to [46]. A fuel cell has high efficiencies at low power [37], so fuel cells scaled in five different sizes between the two conditions will be studied:

• The maximum fuel cell stack power is equal to the maximum power required by the mission profile, and the rated power is equal to the cruising power.

  $${\varphi }_1={\displaystyle \frac{P_{scaled}^{max}}{P_{original}^{max}}}={\displaystyle \frac{4.1\mathrm{MW}}{400\mathrm{kW}}}=10.3$$

  (18)
• The rated fuel cell stack power equals the maximum power required by the mission profile.

  $${\varphi }_5={\displaystyle \frac{P_{scaled}^{nom}}{P_{original}^{nom}}}={\displaystyle \frac{4.1\mathrm{MW}}{243\mathrm{kW}}}=16.9$$

  (19)

The scaling factors are $\varphi =\left[{\varphi }_1{\varphi }_2{\varphi }_3{\varphi }_4{\varphi }_5\right]=\left[10.311131516.9\right]$, arbitrarily chosen between the two boundary conditions. The polarization curves of the five stacks (Figure 6) show how the voltage increases as the number of cells in series increases. The parameters of the five different fuel cells are provided in Appendix C. When scaling the fuel cell stack, a proportional increase in the device’s weight was considered; however, this increase is negligible compared to the aircraft’s total weight, which remains constant across the five different case studies, as does the power demand in the mission profile.

In the study, new fictitious fuel cell stacks are modeled and simulated using OpenModelica. The simulation employs the DASSL solver, an advanced extension of Newton’s method that uses higher-order derivative approximations [54].

The power required by the mission profile is always achieved in the five simulations, and it is observed that oversizing a fuel cell results in higher efficiencies in both operational phases considered. Figure 7a shows the heat produced by the fuel cell stack calculated by the detailed method. The graph shows the results for two flight phases: at takeoff, where the required electrical power is highest and depicted in the figure, and during cruise, where the required power settles around 2.5 MW. It can be seen that, during takeoff, only for very oversized fuel cell stacks is there less heat loss than the electrical power, while, at lower power during cruise, the heat produced is always less than the power delivered. This is reflected in the efficiency, which, as the FC size increases (Figure 7b), it increases due to the lower operating currents. The efficiency is calculated with the relation described by Hartmann et al. [39]:

$$\eta ={\displaystyle \frac{2F{V}_{fc}}{N\mathsf{\Delta }{h}_{HHV}}}$$

(20)

Therefore, this model enables considerations regarding designing electrical and cooling system architectures as the fuel cell size changes. It can be concluded that an oversized fuel cell makes it possible to reduce the heat produced by decreasing the size and weight of the cooling system. In addition, as shown in Figure 8, hydrogen consumption and water production are reduced by increasing the stack efficiency, reducing the mass of hydrogen needed, and thus decreasing the size of the tanks. However, a higher voltage leads to a higher mass of the fuel cell and some systems in the BoP.

4. Discussion

This paper presents a mathematical model in the OpenModelica environment to simulate fuel cell system architectures, specifically targeting scalability and flexibility for future aircraft with hydrogen-based power systems. The model successfully achieved most of the set objectives, including developing a scalable PEMFC model, validation against real data, and the creation of a basic BOP model. However, some limitations remain for the BOP, lacking a cooling system, thus affecting the system simulations’ accuracy.

Despite this limitation, the current work offers a flexible model for the early stages of aircraft system design, providing reliable simulations of the PEMFCs, although its accuracy is largely dependent on precise parameter selection. The case study demonstrates that the model can aid in defining the fuel cell stack configuration by calculating numerous performance parameters.

Future work will focus on three key areas: improving the accuracy of the PEMFC model, enhancing the BoP component models, and adapting the model to account for SOFC behavior. The model accuracy can be enhanced by incorporating feedback on the stack temperature to flatten the polarization curve further and eliminate one of the model’s underlying assumptions. Additionally, gas flow modeling can be optimized for high-temperature PEMFC applications. Refining the BoP components involves including a scalable cooling system model to better assess the performance under high-power conditions. Furthermore, future studies will explore the potential for utilizing generated heat to improve system efficiency, as well as the implementation of control strategies for various BoP components. Lastly, the model can be adapted to support SOFC studies by modifying the gas channel system.