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Current Issue: 2020  Archive: 2019
Open Access Original Research
A Simplified Optimization Model for Sizing Proton-Exchange Membrane Fuel Cells

Kosmas A. Kavadias *, Stefanos Tzelepis 

University of West Attica, Department of Mechanical Engineering, Laboratory of Soft Energy Applications & Environmental Protection, 250, Thivon & P. Ralli Str., Campus Ancient Olive Grove, Athens, Greece

Correspondence: Kosmas A. Kavadias

Academic Editor: Ramon Costa-Castello

Special Issue: Modeling and Control of Fuel Cell Systems

Received: June 03, 2020 | Accepted: July 12, 2020 | Published: July 24, 2020

Journal of Energy and Power Technology 2020, Volume 2, Issue 3, doi:10.21926/jept.2003011

Recommended citation: Kavadias KA, Tzelepis S. A Simplified Optimization Model for Sizing Proton-Exchange Membrane Fuel Cells. Journal of Energy and Power Technology 2020;2(3):24; doi:10.21926/jept.2003011.

© 2020 by the authors. This is an open access article distributed under the conditions of the Creative Commons by Attribution License, which permits unrestricted use, distribution, and reproduction in any medium or format, provided the original work is correctly cited.


The fluctuations in the cost of energy coupled with the gradual decrease in natural resources (i.e., oil, natural gas, coal) and the environmental issues caused by the extensive use of fossil fuels, demand the urgent need for the development of advanced and clean energy systems. The European Union has requested new measures to target climate and energy deterioration by developing renewable energy sources by 2030. The stochastic character of the energy sources and the fluctuation in demand for the energy systems poses a hindrance to achieving these targets. In this direction, hydrogen technology, that can be produced by electrolysis carried out in the presence of electricity from renewable energy systems can contribute as an energy carrier. It does not emit hazardous gasses and can be stored in the metal hydride canisters for future use via fuel cells (FCs). Modeling these systems is important for achieving their optimal size and to increase the penetration of renewable energy. They are beneficial both from the energy and economic point of view. For the accurate understanding of the hydrogen systems and especially FCs, it is important to understand the thermodynamic and electrochemical operating principles. In the present study, an integrated mathematical model is proposed concerning the electrical and thermal behavior of the FC. The model developed can simulate the operation of a FC by using the FC’s technical specifications. The reliability of the model was validated under three different hydrogen flow patterns applied in the experimental configuration (NEXA 1.2 kW). The proposed model was applied to a specific case study for the optimum sizing of a FC in terms of maximum hydrogen absorption. A hydrogen production pattern from electrolysis was used, and the excess energy from a wind park installed in an autonomous grid island network was utilized.


Fuel cell; optimization; hybrid energy systems; hydrogen; renewable energy sources


1. Introduction

Climate change caused by human activities has raised great concerns and urges for the adoption of policies that will reduce the human footprint on the environment. One of the key players in climate change is the extensive use of fossil fuels to meet the ever-increasing energy needs of the contemporary way of living. New policies have been proposed for identifying new sources and developing new technologies for energy production. At present, researchers are revisiting the technological knowledge gained years ago before the discovery of fossil fuel, to solve the present-day problems.

Renewable energy sources, as an alternative energy solution, are at the forefront of tackling climate change. The extensive use of renewable energy has raised the problems in grid integration primarily caused due to the intermittent nature of the source [1]. Moreover, the easy availability of the source calls for distributed power generation systems, which further increases the concerns for the gaps in the production and consumption demands [2]. The major applications of hybrid systems can be found in the isolated and remote areas, far from the reach of electricity grids [3]. A challenge in standalone systems concerns their optimum sizing for maximum load satisfaction and the minimization of the energy needs.

To address these challenges, hybrid energy systems from different sources and solutions for storage difficulties have been extensively investigated [4]. These systems incorporate different storage solutions, batteries being the most common devices (in small to medium scale systems). Researchers have restarted work on FCs that promote hydrogen as an alternative fuel [5]. However, the production of hydrogen remains a challenge since this is a fuel that does not exist in nature in its pure form. If one considers electrolysis for producing hydrogen, powered by renewable energy, and coupled with FCs, then a storage solution is obtained through the electricity-hydrogen-electricity cycle, as described in Figure 1. Excess hydrogen can be directly used as a fuel at different types of consumption systems (e.g., hydrogen cars).

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Figure 1 Renewable energy sources/Electrolysis/FC configuration.

Hybrid renewable energy systems combined with hydrogen storage can be promising solutions given the flexibility of using hydrogen. Proton-exchange membranes are commercial FCs [6] with a power range from a few watts to multiple kilowatts [7]. They can power a range of devices from portable electronic pieces of equipment, automotive applications to domestic electric loads. To reduce the cost and improve the energy consumption of the renewable energy hydrogen systems, all the parameters affecting the operation of each component under a specific energy profile should be taken into consideration while operating as a storage solution in standalone electrical grids. One of the most reliable methods of doing so is simulating the operation of the hydrogen system.

In the last two decades, efforts have been made to model an electrolysis/FC system combined with a renewable energy system and to identify the parameters affecting its operation. Dutton et al. studied the optimum size of a wind-hydrogen system and the effects of fluctuating power supplies in electrolyzer operation [8]. The examined scheme was composed of a 20 kW wind turbine and a 10 kW electrolyzer. Castañeda et al. developed a sizing method along with different control strategies for suitable energy management of a standalone hybrid system. It consisted of a 6 kW PV system, a 1.2 kW proton-exchange membrane FC, a 3 kW electrolyzer, a 2.97 kg hydrogen storage, and 10.5 kWh batteries [9]. Valverde-Isorna et al. presented a methodology for modeling and evaluating the performance of a wind-hydrogen system. They also validated the estimated parameters [10]. Özgirgin et al. modeled a PV module-electrolyzer hydrogen system for developing control strategies, which can reduce harmful emissions. The investigated configuration is composed of PV modules, batteries, FCs, and an electrolyzer [11].

Alavi et al. analyzed the reliability of a standalone wind-hydrogen energy conversion system [12]. Haddad et al. examined the combination of solar energy systems with FCs for reducing the emission of greenhouse gas. Both the systems were analyzed for different parameters affecting their performance, and recommendations for further improvement were suggested [13]. The rationale behind these systems is the development of a configuration consisting of FC’s coupled with renewable energy solutions to constitute green-green1energy systems. Haddad et al. also examined a triple wind-solar-FC configuration to overcome the energy deficit that is prevalent in several months of the year [14]. These results were based on experimental configurations for investigating the behavior of a FC in conjunction with renewable energy. Several research groups have reported the optimal operating conditions by developing simulation models [15,16,17]. Meyer et al. determined the optimal operating point and the overall electro-thermal behavior for a FC by measuring the ‘current of lowest resistance’ using electrochemical impedance spectroscopy [15]. They have also examined the performance of a FC by applying an electro-thermal performance map to establish the optimum rate of airflow. Their studies focused on maximizing the net power output, which consequently affected the overall efficiency [16]. Santa Rosa et al. published an experimental study on the development of a FC and investigated its performance under several operating conditions viz. cell temperature, rate of airflow, and pressure [17]. The existing work on simulating FC is based on experimental studies for the development of a model that takes into account both the thermodynamic and electrochemical parameters. Though the results obtained are quite interesting and remarkable, reproducibility is a major challenge due to the complex procedures that need to be followed.

This work aims to provide a simplified simulation model for air-cooled, open cathode proton-exchange membrane FC developed via the MATLAB software, based on thermodynamic, electrical, and thermal phenomena occurring at each part of the FC. The model combines the electrical and thermal phenomena and employs the semi-empirical equations developed by different research groups. The developed model aims to simulate the operation of the FC and provide all the necessary information for determining the optimum size as part of an integrated hybrid renewable energy solution in an isolated grid. The factors kept in mind were minimum energy production cost and maximum renewable energy penetration. Research on simulating the hydrogen storage solution in isolated grids incorporating renewable energy was conducted, where initially, a simulation model of the electrolysis was developed [18].

2. Methodology

The proton-exchange membrane is a hydrogen FC that converts the chemical energy of the fuel into a direct current [19]. This type of unit cell is composed of 2 electrodes, one positive (the cathode) and one negative (the anode). The operating principle is based on the reaction of hydrogen and oxygen to produce electricity. Initially, separation of hydrogen into electrons and positive ions occurs at the anode’s electrode-electrolyte interface. Subsequently, the electrons are released and transferred through an external electrical circuit to be recombined at the cathode. The cell has a solid electrolyte where the positive ions are conducted and transported between electrodes. The electrolyte should prevent electrons from passing through the electrolyte to avoid disorganization of the electrochemical reactions, which can lead to a decrease in the FC’s overall efficiency. Usually, the catalyst is noble metal platinum, a key factor for providing stability in terms of the rate of electrochemical reactions. The electrodes are composed of a thin layer of catalyst particles (typically platinum as mentioned) supported by carbon compressed between the ionomer membrane and the porous gas diffusion layer. The structure of electrodes is porous, which helps in the easy transport process of the reactants.

The inlet hydrogen undergoes an electrochemical half-reaction, the Hydrogen Oxidation Reaction (HOR), where hydrogen is dissociated in positive ions (i.e., protons) and electrons. Subsequently, the electrons are released at the anode. At the same time, protons are conducted by the electrolyte and transferred to the cathode. At the cathode, oxygen undergoes an electrochemical half-reaction, the Oxygen Reduction Reaction (ORR) with the incoming oxygen, electrons, and the positive hydrogen ions. The ORR produces water and releases heat [5].

2.1 Basic Chemistry

The basic electrochemical half-reactions take place simultaneously at the anode and the cathode. Hydrogen is separated at the anode into protons and electrons by the oxidation reaction that occurs at the catalyst layer. This is expressed as:


In the catalyst layer at the cathode, oxygen combines with protons and electrons, coming simultaneously from the anode and the external circuit. The reduction reaction is expressed as:


The total reaction is given by:


It is essential to comprehend the fundamental thermodynamic properties that occur during the operation of FC before understanding the two major submodels (electrical and thermal).

2.2 Theoretical Electrical Work

The portion of the energy (higher heating value of hydrogen) that can be converted into electricity represents the maximum amount of thermal energy that can be extracted from hydrogen during the electrochemical reaction. This is the Gibbs free energy or the theoretical electrical work of the FC and is calculated as [20]:


Here, Z denotes the number of electrons per molecule of H2 exchange (Z = 2), F is the Faraday constant describing the electric charge of one mole of electrons in the units of C/mol and E0 is the theoretical FC potential in V. The energy is calculated as the difference between the heat of formation of products and reactants (enthalpy of the chemical reaction):

$$\Delta H\ = \left( \text{h}_{\text{f}} \right)_{\text{H}_{\text{2}}\text{O}}\text{-}\left( \text{h}_{\text{f}} \right)_{\text{H}_{\text{2}}}\text{-}{{^1}/{_2}\left( \text{h}_{\text{f}} \right)}_{\text{O}_{\text{2}}}\tag{5}$$

The enthalpy of formation of an element is equal to zero. However, the heat of formation of liquid water at standard conditions (ΔH) equals -286.02 kJ/mol, where the negative sign indicates the release of heat. From eq. 4, the Gibbs free energy is therefore calculated as:

$$\Delta G\ = \Delta H - T \cdot \Delta S\tag{6}$$

where ΔH is the variation in enthalpy between the heat of formation of the products and the reactants in kJ/mol, ΔS is the variation in entropy between the products and reactants in kJ/(mol·K), and T is the temperature of the FC in K.

Similarly, the entropy, expressed as the difference between the entropy of products and reactants, is calculated as:

$${\Delta S\ = \left( \text{s}_{\text{f}} \right)}_{\text{H}_{\text{2}}\text{O}}\text{-}\left( \text{s}_{\text{f}} \right)_{\text{H}_{\text{2}}}\text{-}{{^1}/{_2}\left( \text{s}_{\text{f}} \right)}_{\text{O}_{\text{2}}}\tag{7}$$

The values of the enthalpy and the entropy for reactants and products for the standard conditions are given in Table 1 [21].

Table 1 Values of enthalpy and entropy of reactants and products.

By substituting the values given in Table 1 in eq. 6 and eq. 7, we get the variation in the entropy equaling -0.163285 kJ/(mol·K). The Gibbs free energy equals to -273.341 kJ/mol.

2.3 Theoretical FC Potential

The reversible cell potential Erev is calculated from a modified form of the Nernst equation where an extra term is added to account for the changes in reversible voltage at different temperatures deviating from the reference temperature [22]:

$$\text{E}_{\text{rev}}\text{=}\frac{\text{ΔG}}{\text{2F}}\text{+}\frac{\text{ΔS}}{\text{2F}}\text{·(T-}\text{T}_{\text{a}}\text{)+}\frac{\text{R}_{\text{g}}\text{·T}}{\text{2·F}}\text{ln}\left\lbrack \text{p}_{\text{H}_{\text{2}}}\text{·}\text{p}_{\text{O}_{\text{2}}}^{\text{0.5}} \right\rbrack\tag{8}$$

Here, Rg is the universal gas constant in J/(mol·K),$\ \text{p}_{\text{Η}_{\text{2}}}$ and $\text{p}_{\text{O}_{\text{2}}}$ are the partial pressures of the corresponding reactant gases in atm, T denotes the temperature of the cell, and Ta corresponds to the ambient temperature in K.

2.4 Calculation of Molar Flow Rates

There is a relationship that links the current delivered by each cell, the flow rate, the consumption of the reactants, and the corresponding production of water and heat [21]. Αn air-cooled, open cathode FC stack is designed to consume the oxygen of the forced air supplied in the cathode while the anode is supplied with hydrogen. Figure 2 illustrates the inputs and outputs.

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Figure 2 Inputs/Outputs flow.

According to the half-reaction at the cathode, the molar flow rate (in mol/sec) of oxygen as a function of current is calculated as:


where I denotes the current A, and NS gives the total number of cells. Similarly, the flow rate of hydrogen consumed at the anode is determined by:


The molar mass of $\text{H}_{\text{2}}$,$\text{O}_{\text{2}}$ and the corresponding utilization factor has to be taken into consideration for estimating the hydrogen and oxygen flow rates in kg/sec (eq. 11 and 12). The utilization factor U is defined as the ratio of the total amount of reactant gases entering the FC to the amount of reactant gases consumed. According to researchers, FC presents stable operation at the fuel utilization values higher than 96% [23]. The present study considers a utilization factor for both hydrogen and oxygen constant at 96%.

$${\dot{\text{m}}}_{\text{H}_{\text{2}}}\text{=}\frac{\text{I·}\text{N}_{\text{s}}}{\text{Z·F}}\text{·}\frac{\text{PM}_{\text{H}_{\text{2}}}}{\text{U}_{\text{H}_{\text{2}}}}\tag{11}$$ $${\dot{\text{m}}}_{\text{O}_{\text{2}}}\text{=}\frac{{\dot{\text{m}}}_{\text{H}_{\text{2}}}}{\text{PM}_{\text{H}_{\text{2}}}\text{·Z}}\text{·}\frac{\text{PM}_{\text{O}_{\text{2}}}}{\text{U}_{\text{O}_{\text{2}}}}\tag{12}$$

where PM is the molar mass in gr/mol (for H2 = 2.01 and O2 = 32).

The current produced is calculated by eq. 11 as:


The overall performance of the FC is evaluated by combining the electrical and the thermal model governing its operation.

2.5 Electrical Submodel

By connecting FCs to an external electrical circuit, the actual electrical potential is much lower than the theoretical value. Since voltage loss is directly associated with the performance of a FC, its voltage output should be calculated, taking into account the voltage loss occurring during the operation of the FC. This is expressed as:


where Erev is the reversible FC potential in V while Vact, Vohm, and Vconc refer to the voltage drop of the FC due to the loss in activation, ohmic, and concentration respectively in V.

Activation Losses are associated with the sluggish kinetics of the electrochemical reactions on the electrodes at both sides of the membrane. This produces a non-linear voltage drop (activation polarization). To be precise, the activation polarization is greater at the cathode due to the kinetics of oxygen reduction, which is slower than the oxidation of hydrogen at the anode. A simplified way to calculate the activation losses is to use the empirical Tafel’s equation [24]:

$$\text{V}_{\text{act}}\text{=}\frac{\text{R}_{\text{g}}\text{·T}}{\text{a}\text{·z·}\text{F}}\text{ln}\left( \frac{\text{i+}\text{i}_{\text{n}}}{\text{i}_{\text{0}}} \right)\tag{15}$$

where a is the charge transfer coefficient expressing how the change in the electrical potential across the reaction interface changes the reaction rate. The charge transfer coefficient can also be considered as a symmetry coefficient. i represents the current density in A/cm2, in is the equivalent internal current density in A/cm2 and i0 denotes the exchange current density in A/cm2 representing the rate at which the reaction proceeds in both the directions at the reference temperature and pressure.

Concentration Losses correspond to the voltage reduction associated with the mass transport of the reactant gases. The mass transport phenomena occur by both convection and diffusion during the operation of a FC. Convection is associated with the transport of reactants due to the bulk movement and diffusion indicates the transport of reactants in the electrodes of a FC [25]. The concentration voltage is calculated by taking into account the following empirical equation:


where m and n are constants (in V and cm2/mA respectively) obtained by a non-linear regression analysis [26]. The mass transport overpotential as a function of current density was taken into account in this study.

Voltage drop due to Ohmic Losses are associated with the resistance to the flow of ions through the electrolyte and the flow of electrons through the electrically conductive components of the FC system. The ohmic voltage is calculated according to Ohm’s law [25]:


where Rohm refers to the ionic resistance. The theoretical values of the ionic resistance range between 0.1 Ω·cm2 and 0.2 Ω·cm2 [25]. However, experimental studies have shown that the resistance of the electrolyte during operation ranges between 0.18 Ω·cm2 and 0.28 Ω·cm2 [27,28]. Biichi et al. [27], who investigated the ionic resistance of a Nafion membrane for current densities between 0-0.777 A/cm2 with a cell operating temperature at 60 °C, reported that the ionic resistance of the membrane falls in the range of 0.18-0.22 Ω·cm2. The experimental study conducted by Springer for a FC operating at 80 °C, for current densities between 0.1 A/cm2 and 0.8 A/cm2, showed that the membrane’s ionic resistance ranges within 0.18 Ω·cm2 - 0.28 Ω·cm2[28]. Considering the above, in the present study, Rohm is taken as 0.2 Ω·cm2 to match the real-life operating conditions. Nevertheless, it has a negligible effect on the overall accuracy of the calculations.

2.6 Thermal Submodel

The thermal behavior of FCs has a vital role in calculating the thermal energy produced due to the electrochemical reactions at the anode and the cathode. A FC converts a part of the internal energy contained in hydrogen into electricity. The irreversible reaction contributes ~55%, the entropic heat (the system’s overall thermal energy per unit temperature (J/K) contributes ~35%, the heat produced by the Joule effect due to the ohmic resistance of the electrolyte contributes ~10% of the total heat released. Furthermore, heat transfer processes in a FC varies at the different components. Thermal conduction occurs due to heat transfer through the membrane, while conduction and convection take place simultaneously in the catalyst layer (CL) and the gas diffusion layer (GDL) [29]. The thermal energy balance of a FC operating at a temperature lower than the maximum limit (the auxiliary cooling system is not operating), can be expressed as [24,30]:



$${\dot{\text{Q}}}_{\text{chem}}\text{=}{\dot{\text{n}}}_{\text{H}_{\text{2}}}\text{·ΔG}\tag{19}$$ $${\dot{\text{Q}}}_{\text{elec}}\text{=i·}\text{V}_{\text{cell}}\tag{20}$$ $${\dot{\text{Q}}}_{\text{loss}}\text{=}\frac{\text{1}}{\text{R}_{\text{t}}}\text{·}\left( \text{T-}\text{T}_{\text{a}} \right)\tag{21}$$

Here, ${\dot{\text{Q}}}_{\text{chem}}$ is the heat produced due to the electrochemical reaction in W, ${\dot{\text{Q}}}_{\text{elec}}$ is the electrical power generated by the FC in W, ${\dot{\text{Q}}}_{\text{loss}}$ denotes the heat transfer loss to the environment in W, $\text{C}_{\text{t}}$ indicates the overall thermal capacity of the stack in J/K, ${\dot{\text{n}}}_{\text{H}_{\text{2}}}$ represents the hydrogen flow rate calculated by eq. 11, Vcell is the FC’s voltage output considering voltage loss in V, and Rt is the overall thermal resistance2 of the electrolyte in K/W.

By assuming a constant heat generation and a constant rate of heat transfer for each time interval, the FC’s temperature can be expressed as:


where $\text{T}_{\text{(t-1)}}$ is the FC’s operating temperature at period t-1. T is the operating temperature for the examined period t, in K.

If the FC’s operating temperature reaches the maximum operating limit, the cooling system takes over to maintain the constant temperature at the operating limit. The auxiliary cooling system operates for meeting the temperature requirements. During the operation of the FC after reaching the maximum operating temperature, the term $\text{C}_{\text{t}}\frac{\text{dT}}{\text{dt}}$(net heat flow stored in the FC) remains zero (dT = 0). In this case, the amount of thermal energy that needs to be rejected via the cooling system is calculated as:


Based on the above calculation procedure, an integrated calculation model was developed to optimally size the FC systems for available hydrogen flow rate profiles. The steps employed for simulating the FC operation has been presented as a flowchart in Figure 3. The model is based on four sets of equations where each has a vital role to play in the FC’s operation under specific operating conditions. The first section covers the procedure related to the amount of reactant gases that enter the FC and the calculation of hydrogen and oxygen flow rates. The second section of the model deals with the calculation of the current produced, which is highly dependent on the amount of gases reacting at the anode and the cathode catalyst layers. The third section involves the calculation of voltage drop due to the electrochemical phenomena occurring in the FC. The fourth and the last section governs the calculation of the Thermal Losses and how the FC’s operating temperature gets affected during the aforementioned electrochemical reactions and transfer phenomena.

The input data required for simulating the operation of FC includes the number of cells, the area of the electrodes, the range of the operating temperature, and the operating pressure. Based on the technical specifications of the simulated FC, the algorithm calculates the electrical power produced for a given hydrogen flow. The simulation procedure takes into account the operating limits of the lower and upper hydrogen flow, as determined by the manufacturer. According to the calculation procedure, the FC absorbs the amount of hydrogen within its operating limits. The algorithm takes into account the said variables and calculates the FC stack’s voltage, current, and the total power output according to the steps outlined in Figure 3.

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Figure 3 Calculation procedure.

3. Model Reliability Test

To validate the accuracy of the developed model, the algorithm was tested and compared with the Nexa Training System from the Soft Energy Applications & Environmental Protection Lab [31]. The Nexa Training System is an experimental configuration consisting of a 1.2 kW FC, an electronic load, battery modules, a power management module, and three metal hydride canisters for hydrogen storage. The technical data of the system are presented in Table 2 [32].

Table 2 Nexa 1.2 kW technical datasheet.

The Nexa FC was operated under three different consumption profiles, which were selected based on the capability of the system. Figure 4 depicts the profiles used during the validation of the experimental procedure. The profiles represent the various hydrogen flow rate patterns absorbed from the FC system. Scenario A indicates that the hydrogen supply increases every 20 sec till it reaches ̴14 Nl/min where it remains constant for 20 sec and then drops to zero. In scenario B, the FC initiates at 8 Nl/min, and after 15 sec it starts increasing linearly till it reaches 12 Nl/min for 50 sec, and subsequently drops to 5 Nl/min for 10 sec, and thereafter the value rapidly decreases. In scenario C, the hydrogen supply increases linearly up to 14 Nl/min and then rapidly drops to zero. The investigated consumption profiles could reflect the FC’s operation as part of an integrated hybrid-renewable energy configuration. The hydrogen delivered in FC could be produced by electrolysis, utilizing the surplus energy from renewable energy systems, covering the fluctuations in demand. The electrical energy produced by the renewable energy systems can be converted to hydrogen via electrolysis and stored, in the form of hydrogen, in metal hydride canisters. If the energy demand exceeds production from the renewable energy system, the FC absorbs the stored hydrogen and generates electricity to cover the demand deficit.

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Figure 4 Investigated consumption scenarios.

The technical characteristics of the Nexa FC provided by the manufacturer, along with the three different hydrogen profiles, were used as input data for testing the reliability of the developed algorithm. Before proceeding with the validation of the voltage and current parameters for the three investigated scenarios, the power curve of Nexa FC was reproduced by the developed model based on the technical specifications given in Table 2. A comparison between the calculated power curve and the power curve provided by the manufacturer is presented in Figure 5. The two power curves are similar, verifying the accuracy of the proposed algorithm.

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Figure 5 Model reliability test on Nexa FC power curve

The algorithm predicts the current values with higher accuracy than the voltage in all the cases (Figure 6), as explained by the association of voltage with loss that cannot be accurately predicted. More precisely, the calculation of the voltage is based on empirical and semi-empirical equations where the thermodynamic and electrochemical phenomena are simplified to a large extend. This leads to an overestimation of the calculated voltage value. The algorithm underestimates the current values with the highest differences appearing at high operating points of the FC attributed to the presence of higher current density and the higher ratio of hydrogen utilization factor, a parameter which is considered constant in the model. The statistical evaluation results (Table 3) include basic statistical indices, i.e., the coefficient of determination (R2), the Mean Bias Error (MBE), and the relative Mean Bias Error (MBE%). The mean bias error between the calculated values (O) and measurements (M) of a sample consisting of n values is defined as the measure of the overall bias error or systematic error. This is expressed as:

$$\text{MBE}\text{=}\frac{\text{1}}{\text{n}}\sum_{\text{i}\text{= }\text{1}}^{\text{n}}\text{(O-M)}\tag{24}$$

The relative mean bias error of n values is expressed as:

$$\text{rMBE =}\frac{\text{MBE}}{\frac{\text{1}}{\text{n}}\sum_{\text{i = 1}}^{\text{n}}\text{O}}\tag{25}$$

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Figure 6 Comparison of FC stack voltage and current between experimental and calculated values for the three scenarios under investigation.

Table 3 Statistical indices.

The results show that the correlation between the observed and the predicted values is significantly high. The coefficient of determination was in most cases higher than 90% while the voltage MBE ranges between 5.6 V - 6.8 V and the current MBE ranges between 1.5 A - 5 A. The rMBE falls in the range of 20.40% - 28.84% for the voltage and within -16.08% to -8.55% for the current. The overestimation of the voltage output and the corresponding underestimation of the current can be seen from the positive and negative signs in the MBE and rMBE indicators. For the power output curve, the R2 presents high accuracy (99%), while the MBE and rMBE are equal to 36.68 W and 4.64%, respectively (Figure 5). By comparing the model output with the experimentally obtained data, correction factors could be introduced in the model to improve the calculation procedure. Such modifications in the model require a significant number of measurements at different FC modules, which have not been considered in the present work but will be reported in subsequent publications.

For further validation of the proposed model, the experimental and the simulation data were compared with the results from similar studies. Haji et al. [33] investigated the performance of a 40 W FC consisting of 10 cells connected in series with an electrode with a surface area of 25 cm2 and the supplied hydrogen pressure at ̴1.6 bar. The experimental data obtained were used for calculating the parameters of a complete analytical model that describes the polarization curve. The Levenberg-Marquardt algorithm was applied for determining the values of the parameters by minimizing the sum of squares of the error. Laurencelle et al. [34] investigated the performance of PGS-105B FC composed of 36 cells of 232 cm2 active area each. The supplied hydrogen pressure was regulated to 3 atm. The study estimated the polarization curve by using the polarization curve-fitting equations.

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Figure 7 Comparison of polarization curve with similar studies.

There is a good agreement between the proposed model and the compared studies (Figure 7). The proposed model presents a better estimation of the polarization curve as compared to the reports of Haji et al. since a statistical method was applied in the experimental study in which the observed data are modeled by a non-linear combination of the model parameters. The proposed model presents a better agreement to the manufacturer’s polarization curve compared to the results obtained by Laurencelle et al. These are attributed to the fact that their model did not consider the essential parameters that indicate the rate of electrochemical reactions, e.g., the internal current density and exchange current density. Thus, the developed model calculates the manufacturer’s polarization curve with better accuracy.

4. Case Study

The developed algorithm was applied to investigate the operation of a FC system combined with renewable energy and electrolysis. The rationale behind this was to utilize the surplus energy rejected by renewable energy systems in an electrolysis system for the generation of hydrogen, which can either be consumed immediately using FCs or stored in metal hydride canisters for future use. In both cases, the hydrogen flow pattern absorbed by the FC system depends on the electrical loads. A FC system is examined in terms of optimal size to fulfill the preconditions of maximum hydrogen absorbance3 capability of a certain hydrogen supply profile. The scheme under consideration is composed of a wind farm consisting of 5 wind turbines rated at 2 MW each, combined with a 1 MW hydrogen generator and a FC system.

Figure 8 indicates the hourly hydrogen profile produced by the electrolysis system during the first 2000 h of operation. The operating limits of the electrolysis system range between 2 kg/h - 20 kg/h. The present study aims to determine the FC’s optimal size to ensure maximum absorption of the hydrogen produced and to attain an increase in the absorption efficiency of the electricity-hydrogen-electricity cycle.

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Figure 8 Hydrogen flow rate pattern.

The parameters for the simulation process are based on the typical values of a FC (Table 4). The study was carried out at an operational pressure of 2.5 bar, with cells connected in series for avoiding high operating current values. The value of the internal current in ranges between 0.5 A/cm2 - 1.5 A/cm2 [35]. The exchange current density varies between 0.1 A/cm2 - 0.9 A/cm2 [36].

Table 4 FC parameters used for the simulation.

The various FC sizes were simulated by varying key parameters such as the number of cells and the surface of the electrode. One of the examined cases, presented in Figure 9, concerns a FC configuration rated at 300 kW. Figure 9a shows the power curve as a function of the current and Figure 9b presents the corresponding polarization curve (V-I) for the examined FC.

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Figure 9 Power curve (a) and polarization curve (b) of the examined FC.

The aforementioned configuration (300 kW FC) was examined in terms of hydrogen absorbance capabilities. Figure 10 presents the relation between the hourly hydrogen absorbance with the available profile produced by electrolysis, while the maximum and minimum limits of flow rates range between 3 kg/h - 18 kg/h. The FC absorbs hydrogen within its operating limits according to the management applied by the algorithm.

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Figure 10 Hydrogen absorbance.

Τhe FC temperature affecting the FC’s thermal behavior is calculated at each time step during the variation of the operating conditions. According to the technical specifications, the maximum temperature was set at 70 °C. The cooling system operates above this limit to regulate the temperature. Figure 11 depicts the variation in the operating temperature along with the power output for the examined period. The presented period is the one with the highest fluctuations in the operating temperature of the FC. According to Figure 11, the FC’s temperature ranges between 60 - 70 °C for most of the period under investigation. The higher temperature drop presented within this period is due to the reduction of the FC’s operating load as a result of the significant low electrical loads that need to be covered.

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Figure 11 Temperature fluctuations.

The annual energy balance of the FC is distributed between the electrical energy (Qelec) produced and the Heat Losses generated due to the thermal energy removed by the auxiliary cooling system (Qcool) as well as the thermal energy emitted to the environment (Qloss). Of the absorbed hydrogen energy, 53% was converted to electricity (Figure 12).

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Figure 12 Annual energy balance.

The optimum sizing of the FC in terms of maximum hydrogen absorptance is demonstrated in Figure 13, which illustrates the results of simulating 600 different FC configurations. According to the results, the hydrogen absorbance increases by ~0.5%/kW for FC configurations up to the maximum absorptance value and decreases by ~0.2%/kW after that. The maximum absorbance achieved, based on the specific profile, is 96.5% of the available hydrogen with a FC configuration of 300 kW.

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Figure 13 Percentage of hydrogen absorbance at different values of FC capacity.

By increasing the capacity of the FC, the corresponding initial, maintenance, and operational costs are also increased. Further investigation of the economic behavior of the different sized FC systems should be considered. A rough estimation reveals that the initial FC cost will be around 1630 Euro/kW [38]. A cost-efficient solution is achieved by a FC with a capacity of 250 kW. Hydrogen absorbance is estimated at 95.8%. The proposed model, combined with an electrolysis simulation model, can be used for estimating the overall energy absorption ratio for a full cycle of renewable energy-electrolysis-FC. In this respect, an electrolysis system in conjunction with renewable energy and a storage system has already been investigated [18]. A combination of these two models can contribute toward the optimum sizing of an electrolysis-FC system, which can be used as a storage medium in hybrid renewable energy systems.

5. Conclusions

An integrated algorithm was developed to simulate the electric power production for different hydrogen flow rates in FC. The algorithm can be applied for various hydrogen excess profiles. It can provide results regarding a FC’s operating parameters as well as the optimum size for the maximum hydrogen absorption. This is a flexible model developed for simulating FC stacks with different technical specifications.

The algorithm was validated by using an experimental unit (Nexa 1.2 kW) and was compared with similar studies. According to the results, the correlation between the observed and predicted values of the current, voltage, and power output parameters is of high accuracy. The values of the coefficient of determination in most cases were higher than 90%, while the MBE ranged between 1.5 - 6 for the current and 5.6 - 6.8 for the voltage. The relative Mean Bias Error lies within 20.40% - 28.84% for voltage and -8.55% to -6.08% for current. The proposed model presented a better approach for the actual voltage-current curve of the fuel cells when compared to previously reported similar studies.

According to the results obtained for the integrated autonomous hybrid system, it can be concluded that a considerable amount of the available hydrogen can be absorbed ( ̴96.5%) whereas the mean energy efficiency of the FC was estimated at 53%. The economic efficiency of the FC system should also be considered before reaching a conclusion since the initial, maintenance, and operational costs increase with size.

The proposed model can be applied for investigating both the operation and the optimum size of a FC configuration as part of an integrated solution which combines renewable energy and hydrogen technologies for maximizing the absorbance of the available hydrogen, at the same time improving the system’s overall reliability by testing different sizes and technical specifications in the FCs. The developed FC model can also be used for simulating FC in vehicles to test the performance and autonomy under variable power demand.

Author Contributions

Authors contributed to the manuscript equally.

Competing Interests

The authors have declared that no competing interests exist.


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