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This paper reports on the modelling of a commercial 1.2 kW proton exchange membrane fuel cell (PEMFC), based on interrelated electrical and thermal models. The electrical model proposed is based on the integration of the thermodynamic and electrochemical phenomena taking place in the FC whilst the thermal model is established from the FC thermal energy balance. The combination of both models makes it possible to predict the FC voltage, based on the current demanded and the ambient temperature. Furthermore, an experimental characterization is conducted and the parameters for the models associated with the FC electrical and thermal performance are obtained. The models are implemented in Matlab Simulink and validated in a number of operating environments, for steady-state and dynamic modes alike. In turn, the FC models are validated in an actual microgrid operating environment, through the series connection of 4 PEMFC. The simulations of the models precisely and accurately reproduce the FC electrical and thermal performance.

Fuel cells (FC) have received a major boost in recent years, as a result of the growing demand by a number of sectors. For this reason, this technology is rapidly expanding and there are many different research lines associated with the various sectors. There are basically three sectors behind the development of fuel cells: namely, the transport sector [

A wide range of FC technologies are available, which are at different stages of development. Although FCs can be classified into a number of categories, based on the type of fuel used (such as hydrogen, methanol or natural gas), the operating temperature (ranging from ambient temperature up to 1000 °C), FCs are generally classified according to the type of electrolyte. However, regardless of the FC technology used, in all cases the net reaction for the recombination of hydrogen and oxygen to form water, is the same [

The PEMFCs operate at a lower temperature, generally under 100 °C and, therefore, the connection time is faster than for other FC types. In turn, they show a rapid response to load variations, and are also compact, lightweight, noiseless and, furthermore, as they use a solid polymer electrolyte, they are easier to manufacture than other FC types, such as the AFCs. The PEMFCs have been validated in a number of applications such as automobiles, buses, distributed generation, cogeneration, stand-alone systems and portable systems [

The scientific literature contains a number of papers on FC modelling, including theoretical and empirical models, some of which model the FC whilst others are focused on the different FC components such as electrodes, membrane,

The modelling of the FC electrical performance can be classified according to the FC operating mode. Some authors propose steady-state models which analyse the operation of the FC at either a fixed operating point or with slow dynamics [

On the other hand, a number of authors have worked on the FC thermal modelling, proposing semi-empirical models based on thermal balance that are valid for steady-state and dynamic modes [

With regard to the modelling of the various FC components, some authors have developed in-depth models focused on phenomena occurring in the said components, such as the transport of water [

This article reports on the modelling of a commercial 1.2 kW PEMFC configured through an electrical model and a thermal model, able to represent its performance in any operating regime. Firstly, an electrical model is proposed, based on the integration of the different thermodynamic and electrochemical phenomena taking place in the FC. This model predicts the FC voltage for a steady-state and dynamic mode alike. A thermal model is then developed, based on the FC thermal energy balance, capable of predicting the FC operating temperature in relation to the ambient temperature. Once the models have been established, the next step is to characterise and obtain the parameters associated with the electrical and thermal performance. Then the models representing the FC electrical and thermal performance are implemented in Matlab Simulink and are validated in a number of steady-state and dynamic operating environments. Finally, the models are validated in an actual operating environment through the integration of four PEMFCs in the microgrid located in the Public University of Navarra (UPNa).

The FC experimental study was conducted at the UPNa Renewable Energies Laboratory. The laboratory, shown in ^{3}. The four PEMFCs are identical, corresponding to model NEXA1200, supplied by Heliocentris (Berlin, Germany). This FC model obtains oxygen from the air and has a power output of 1200 W, with heat and water vapour being its only by-products. ^{2}. According to the manufacturer's specifications the voltage range is 20–36 V and the maximum current is 60 A. The maximum FC operating temperature is 65 °C and it is equipped with a pressure regulator to maintain the hydrogen operating pressure in the stack at around 1.32 bar (absolute pressure). Likewise, each FC has an external hydrogen solenoid valve, a sensor to measure the hydrogen flow rate, an external power supply to power the FC at start-up and software to start and stop the FC and to obtain the principal operating variables (FC CONTROL). In addition, it incorporates a relay and a diode to prevent damage to the system when the FC is connected to the load. The sensor used to measure the hydrogen flow rate is model GSEM C9TS DN00 of red-y smart series made by Vögtlin Instruments (Aesch, Switzerland).

Furthermore, the laboratory is equipped with other equipment to perform the various experiments required for this present work: namely a programmable electronic load (E-LOAD), shown in

This subsection deals with the modelling of one of the FCs described in Section 2. The modelling developed is configured through an electrical model and a thermal model. The combination of both models predicts the FC electrical and thermal performance.

Temperature is a determining factor in the FC electrical performance, the FC electrical variables shown in

In a PEMFC cell, the electrochemical reaction occurs in which hydrogen and oxygen are combined to produce water. By applying the thermodynamic laws to this reaction, and based on the Nernst equation, the reversible voltage of a cell is obtained (_{rev}) [_{H2} and _{O2} are, respectively, the hydrogen and oxygen pressures (bar).

Bearing in mind that the FC stack comprises _{s} series-connected cells, then the FC reversible voltage (_{rev,s}) is obtained by the following equation:

The use of an electrical circuit to model the thermodynamic phenomena associated with the reaction taking place in the FC is performed using a voltage source (_{rev,s}) dependent on temperature (_{H2} and _{O2}), based on

The _{H2} experimentally presents a slight variation in relation to the current demanded from the FCs (_{FC}). Given the fact that this relationship is practically linear, the following equation is proposed in order to relate the _{H2} with the _{FC}:
_{0} and _{1} are empirical parameters.

The activation phenomena are due to the kinetics of the electrochemical reactions taking place in an FC cell. The transfer of the electrical charge between the chemical species and the electrodes involves an energy demand due to the variation of the Gibbs free energy occurring at the different process stages [_{act}). In a PEM type cell, the activation overvoltage is relatively high and, therefore, it is possible to model the phenomena derived from this effect, with sufficient accuracy, through Tafel's equation:
_{g} is the ideal gas constant, F is Faraday's constant, α is the charge transfer coefficient, _{0} is the exchange current and _{act} is the activation current passing through the cell.

Considering that α and _{0} are unknown quantities, and in order to experimentally obtain _{act} _{0},

The activation losses are affected by temperature. The greater the temperature, the lower the _{act} Therefore, the parameters of

In order to obtain the phenomena associated with the fuel cell activation, account should be taken of the fact that the same current is present in each cell and in the stack, given the fact that the cells are series connected (_{act} = _{act,s}). Based on _{act,s}):

The modelling of the phenomena associated with the FC activation is made using a current source (_{act,s}), as shown in

The concentration phenomena associated with the operation of a FC cell are related to mass transport. The mass transport in a FC cell mainly occurs by both processes of convection and diffusion. Convection refers to the transport of species by the bulk movement of a fluid and diffusion refers to the transport of species due to concentration gradients. The mass transport in the electrodes of the FC cell is mainly dominated by diffusion [

The cell concentration overvoltage (_{con}) is related to the concentration current (_{con}) and can be obtained through the following empirical equation:

In order to obtain the FC concentration overvoltage, account is taken of the fact that the same concentration current is present in each cell and in the stack, as the cells are series connected (_{con} = _{con,s}). Based on _{s} the concentration voltage is obtained for the FC (_{con,s}):

In order to model the phenomena associated with the concentration, the electrical model is proposed, represented by a voltage source (_{con,s}) based on

The temperature increase favours the mass transport mechanisms since it improves the diffusivity of the species involved in the reaction [_{con,s} will decrease. In order to take into account this concentration voltage dependence on temperature and maintaining the compromise between the model complexity and accuracy, it is established that parameter

The dynamic phenomena taking place in an FC cell are associated with the double layer effect. This capacitive effect takes place at the electrode-electrolyte interfaces in each cell. The charge transfer occurs during the oxidation and reduction half reactions taking place at the electrode-electrolyte interface of the electrochemical devices, based on the transfer (oxidation) or capture (reduction) of electrons [

The electrical performance of the double layer effect and the associated charge transfer is similar to that of an RC network, comprising a capacitor (_{dl}), termed a double layer capacitor, which models the effect of the accumulation of ionic and electronic charges, and a resistor known as a double layer resistor. The function of the double layer resistor is to model the kinetics of the electrochemical semi-reactions and mass transfer for small variations in the current in relation to a stable operating point. The non-linear variation of the double layer resistor in relation to the current means that it is not valid for modelling the fuel cell transient performance after a certain amplitude [

Given the fact that the FC has _{s} series connected cells, the FC double layer capacitor (_{dl,s}) is obtained from the fuel cell double layer capacitor (_{dl}) through the following equation:

The electrical modelling of the FC double layer effect is based on a capacitor (_{dl,s}) as shown in

The ohmic phenomena are caused by the resistance of the various fuel cell elements to the flow of ions and electrons. The electrical current flow through the cells leads to voltage losses termed ohmic overvoltage, which can be represented by Ohm's Law [_{ohm}) is primarily due to the electrolyte resistance to the ion flow, in addition to the electron flow resistance offered by the electrodes, bipolar plates, current collectors and their corresponding interconnections. _{ohm} is proportional to the electrical current flowing through the cell and, therefore, can be generically represented based on Ohm's law:
_{ohm} is the net ohmic resistance (Ω) of the cell and _{ohm} is the current flowing through the cell.

The FC ohmic overvoltage is obtained by taking into account that _{ohm} is the same as the one flowing through the stack (_{s}) and _{s}:

The FC ohmic resistance (_{ohm,s}) is related to _{ohm} through _{s}:

The electrical modelling of the FC ohmic overvoltage is based on the ohmic resistance (_{ohm,s}) which linearly relates the FC stack current (_{s}) with the voltage, as can be seen in

The influence of temperature on _{ohm} is primarily due to its influence on the membrane resistivity. Although it also affects the cell's solid conductive components. In this respect, a temperature increase would lead to a linear decrease in the value of parameter _{ohm}. This parameter is modelled through a linear function of the FC operating temperature (

With regard to the FC analysed, part of the energy generated by the stack is used to power the FC peripherals described in Section 2. The peripheral energy consumption is modelled by means of a current source connected in parallel with the FC stack, given the fact that the current generated by the stack branches into the peripherals current and the current generated by the FC. _{per}).

The equation for the current source modelling the FC peripherals and which relates the FC current (_{FC}) and the peripherals current (_{per}) is as follows:

In an FC, only a fraction of the internal energy contained in the hydrogen (fuel) can be converted into electricity, the remaining energy is either dissipated or absorbed by the FC, leading to an increase in the FC operating temperature [_{g} is the heating power generated (W) and _{e} is the heating power dissipated (W) and _{n} is the internal heating power acquired by the system (W).

The balance of power making up the FC heat generation is due to the heating power released by the chemical reaction (_{c}),the electrical power generated by the FC (_{s}) and the heating power associated with the sensible and latent heat (_{la+se}) of the reactants (hydrogen and oxygen) and the reaction product (water) [_{g} is represented by the following equation:

_{c} is determined from the quantity of energy per time unit, entering the system. This depends on the enthalpy potential of the FC and the electron transfer rate in the said reaction. Therefore, it is established that:
^{0}_{f,H2O,1} is the enthalpy of liquid water formation (285.84 kJ mol^{−1} at 25 °C and 1 bar).

_{s} is obtained through voltage (_{FC}) and the FC stack current:

Sensible heat _{se} is defined as the heat received by a substance and which causes its temperature to rise, without affecting its state. This heat is directly proportional to its mass, the specific heat and the temperature difference:
_{a} is the ambient temperature (°C), ^{−1}), _{p} is the specific heat (J g^{−1} °C^{−1})

Latent heat (_{la})is defined as the energy required for a substance to change phase, in this case, the water formed in the reaction changes phase (vaporisation heat):
_{v,H2O} is the enthalpy of vaporisation of water (2410 J g^{−1} at 25 °C and 1 bar).

_{la+se} is obtained by applying

In the calculation, it is assumed that all the hydrogen entering the FC causes a reaction. The mass flow rate of oxygen (_{O2}) and water (_{H2O}) is determined from the reaction stoichiometric coefficients, the molecular mass (_{i}_{H2}):

In addition, there is a relationship between the consumption of hydrogen and the current generated by the FC. Therefore, it is possible to estimate the consumption of hydrogen, oxygen and water based on the current generated by the FC. For this, the following expression is proposed:
_{H2} is the hydrogen consumption in NL min^{−1} (1 NL min^{−1} = 0.0015g s^{−1}) and _{f} is an empirical parameter.

The thermal capacity of a body is defined as the ratio between the quantity of heat energy transferred between the said body and its environment in any process, and the temperature change experienced [_{th} is the thermal capacity of the FC (J °C^{−1}) and is obtained from the FC mass (_{s}) and the FC specific heat (_{p,s}) as follows:

The heating power dissipated from the PC through conduction, convection and radiation. However, the influence of the heat transfer by convection is significantly higher than the transfer by conduction and radiation [

The heating power dissipated is defined as the product between the heat transfer coefficient (_{t}) and the difference between the FC operating temperature (_{a}) [

Based on the geometric configuration of the stack, in addition to the coolant air flow path, it is possible to obtain the heat transfer coefficient, which primarily depends on the air flow rate in the stack. In turn, this depends on the FC fan speed (_{fan}). The _{fan} has a minimum value _{fan,min} until the FC exceeds the reference temperature (_{ref}), after which the said speed increases in proportion to the difference between both temperatures. It is established that _{fan} is governed by the following expression:
_{fan} is an empirical constant.

By substituting _{t}), the equivalence of which is determined by the heat transfer coefficient inverse (_{t}), the equation for the FC operating temperature is obtained:

The terms of the _{th}), the heating power generated is represented by current source (_{g}) and the heating power dissipated by variable resistor (_{t}).

The

The test programmed in the electronic load consisted in always maintaining the same 60 A peak-to-peak sine wave component for a 30 s period whilst varying the DC component. This made it possible to obtain

The results obtained can be seen in

The influence of the concentration losses can be clearly seen above 50 A, where the FC voltage trend changes slightly, decreasing with a greater slope. On the other hand, it can be seen that the increased temperature favours the FC electricity supply, due to the fact that, for the same current, the FC output voltage increases. This is basically due to the increased activity of the redox semi-reactions and to the decrease in ionic resistance, leading to a reduction in the activation, ohmic and concentration losses. In addition,

Firstly, the parameters related to the consumption of the peripherals were obtained (_{0}, _{1}, _{2}). The current generated by the FC stack (_{s}) is the sum of the current delivered by the FC (_{FC}) and the current shunted to the peripherals (_{per}) such as the fan, control circuits and purge valve [_{FC}, _{s} and _{per} were measured. Based on these experimental data, specifically with the relationship between _{FC} and _{per} _{0}, _{1} and _{2} shown in

Likewise, the hydrogen pressure (_{H2}) must be obtained in order to obtain the reversible voltage. For this purpose, the above-mentioned tests were used, in which the _{H2} was obtained in relation to the _{FC} in a steady-state operating mode. Parameters _{0} and _{1} were obtained by fitting _{H2} in relation to the _{FC}.

Then the remaining parameters for the model were determined, related to the steady-state operating mode. These include parameter _{ohm} corresponding to the ohmic phenomena [

By substituting

_{s}). Given that the experimental current corresponds with that generated by the FC (_{FC}) it is necessary to obtain _{s}. To do so, the following expression is used, obtained from

The process to obtain the parameters was made by fitting _{ohm} is that corresponding to the value obtained from the dynamic characterisation of the small signal detailed in Subsection 4.2.2. Once parameters _{ohm}, _{0}, _{1}, _{0}, _{1}, _{0} and _{1} for these expressions, modelling the temperature influence on each parameter. The results of the individual experimental fitting of each parameter, based on temperature, in addition to the modelling of the parameters based on their expressions, are shown in

Where the parameters associated with the activation (

A frequency analysis was made in order to determine the experimental characterisation of the electrical performance of the FC in the dynamic operating mode, and to obtain the model parameters related thereto. The EIS technique applied to the FC consisted in drawing from the FC a DC current _{DC} with a small AC sine wave signal, δ

The experimental tests conducted on the FC using the EIS technique, were based on drawing a DC current (_{DC}) of 5, 10 ,15, 20, 25, 30, 35, 40, 45, 50, 55 and 60 A corresponding to a temperature range from 20 to 62 °C. The current perturbation amplitude (δ_{DC} for a frequency range of 0.1 to 1000 Hz.

Following the EIS tests, the FC impedance was obtained, for the selected DC current point, from the small signal experimental amplitudes and phase shifts for voltage (δ_{DC}, used for the tests.

Being small signal, the circuit shown in _{rev,s}. Likewise, given the fact that the experimental tests are based on small signal perturbations, the influence of the activation phenomena and concentration phenomena were modelled using resistors. The performance of the activation resistor and the concentration resistor were considered to be linear for each _{DC} of the impedance spectroscopy. In short, the current source and voltage source for the model in _{dl,s}, which is equal to the sum of the activation and concentration resistors. Therefore, this resistor represents the linearization of the activation and concentration phenomena, and can be calculated from the partial differentials of the activation and concentration voltages in relation to the current, and taking into account that _{act,s} and _{con,s} must be equal to the current at the point at which the small signal analysis is being made, that is, _{DC}.

The determination of the parameters for the electrical model, associated with the dynamic mode, is based on the experimental results obtained in the EIS, with the impedance of the circuit shown in

The parameters _{ohm,s} and _{dl,s} were obtained by fitting the _{ohm} and _{dl} taking _{s} into account, in other words through _{ohm} parameters were obtained for each operating temperature, the fit was made using _{ohm,0} and _{ohm,1} were obtained, modelling the influence of temperature on each parameter. The individual result and the fit, based on the temperature of the said parameters, is shown in _{ohm} could be seen to fall as the temperature increased. This is primarily due to the fact that the membrane conductivity increases with temperature. On the contrary, parameter _{dl} increases slightly at high temperatures, however, in order to simplify the model, it is considered to be independent of temperature (_{dl} = 4.9183 F). With regard to the ohmic resistance, for a temperature of 60 °C, this acquires a value of 0.0012 Ω (0.18 Ω·cm^{2}), similar to those obtained in [

The EIS applied to the FCs makes it possible to characterise the FC dynamic mode, and also principally serves to identify parameters _{ohm} and _{dl}. These values are considered to be independent of the current. This consideration was validated by performing additional EIS tests at different currents, whilst maintaining the temperature constant. Similar values were obtained for both parameters.

The thermal characterisation of the FC is performed through an analysis of the FC operating temperature evolution compared to the current drawn from the FC. For this purpose, a test was conducted during which a DC current was drawn from the FC with values of 10, 20, 30, 40, 50 and 60 A.

The parameters for the thermal model represented in _{f} is shown in

As was seen in Subsection 3.2, when determining the heat dissipated from the system, it is essential to know the air speed and, in turn, the air speed is dependent on the fan speed. In order to determine the fan speed, the test shown in _{ref}), the fan speed increased in proportion to the difference between _{ref}. This figure also shows the fit made for _{fan} obtained for the fit of

Finally, the heat transfer coefficient (_{t}) was obtained, which determines the dissipated heating power [_{t}) of the thermal model represented in _{t} inverse. The _{t} primarily depends on the air speed and, consequently, on the fan speed. For this purpose, _{t} was determined by means of an empirical expression, depending on the fan speed:
_{H}_{t,1}, and _{H}_{t,2} are empirical coefficients.

In order to obtain _{t} the tests shown in _{t}, in addition to the fitting of the same by means of _{fan} increases, so does _{t}. The values obtained for _{t} are similar to those presented in Ref. [

The long-duration validation of the models proposed was conducted by means of the test shown in _{FC}), the FC voltage decreases (_{FC}) and the FC operating temperature increases (_{FC} precisely follows the experimental _{FC}. It can be observed that, initially, the simulated _{FC} is slightly higher than the _{FC} measured, until the current drawn reaches 40 A, when both voltages are superimposed. On the other hand, for currents of more than 50 A, the simulated _{FC} is less than the measured _{FC}. Likewise, it can be observed that the largest deviation between both voltage values is at the time instant equal to 20 minutes, corresponding to a current step of 10 A. At this instant, the difference between both voltages is 0.86 V, corresponding to a relative error of 3.13%. _{a}) for the test corresponding to _{FC} is 20 A, the simulated

The dynamic validation of the models proposed was conducted through a test in which the FC was required to supply a sine wave current of various frequencies and with a peak-to-peak amplitude of 60 A, superimposed on a 30 A DC current. The test started with a frequency of 0.1 Hz and this was gradually increased to 100 Hz. Before conducting the dynamic tests, a current of 30 A was drawn from the FC for sufficient time to allow the operating temperature to stabilise at 52 °C and to remain constant throughout the dynamic test.

It can be seen that, as the frequency increases (from 0.1 to 10 Hz), the area created in each current- voltage ratio starts to increase due to the fact that the double layer capacitance gradually absorbs the current and, therefore, the FC voltage and current gradually become out of phase. As the frequency increases (from 10 to 100 Hz), the current-voltage ratio gradually reduces its slope and starts to reduce its surface area. If the frequency were to be increased to high values, then the current-voltage area would disappear, as the double layer capacitance would be eliminated and the current-voltage relationship would be primarily determined by the FC ohmic resistance. Likewise, it can be observed that the FC model accurately reproduces the FC performance. A mean RMSE of 0.37 V and a mean MAPE of 1.25% was obtained, corresponding to the mean value of the RMSE and MAPE for the three tests.

This section analyses the integration of an FC system into an electric microgrid located at the UPNa.

Furthermore, the microgrid has a management system (PMS) directed at the real time acquisition and control of the microgrid energy flows. In turn, the microgrid is equipped with a number of devices to monitor and measure the electrical and meteorological variables. The microgrid consumption is emulated through a programmable electronic load as described in Section 2. The electronic load emulation programming is based on real electricity consumption data, measured in a family home located close to the UPNa [

The analysis was made using a real power profile for the microgrid, with a duration of 6 hours. _{PV}) and the wind power generated (_{W}) and the power consumed in the home (_{CON}) on the 9th December 2012 from 13:30 until 19:30 hours. This data was measured in the microgrid with a one second sampling time. _{CON} shows significant power variations due to the different loads connected throughout the period. Part of the consumption is practically constant and is primarily due to an electric heater, fridge and to the stand-by mode of a number of electric appliances such as the TV. Likewise, the day time consumption increased at lunch (13:30 h) and supper (19:00 h), primarily due to the connection of electric cooking appliances. _{PV} showed variations caused by the presence of scattered clouds. _{W} is low due to a low wind speed. The difference between the renewable power generated, in other words the sum of _{W} and _{PV}, and the power consumed is assumed by the storage system. In this analysis, the power of the storage system must be assumed by the four PEMFC (_{FC}) described in Section 2.

The DSP was used to program the microgrid storage system power profile of the microgrid into the electronic load. The FC system was subjected to the power profile of the microgrid storage system. _{FC}) shown in _{FC}, leading to considerable fluctuations in the FC voltage. At different time instants, the FC voltage dropped to values of less than 90 V. In addition, _{FC} is slightly higher than the experimental _{FC}. Specifically, the greatest deviation between both voltages is in the time interval from 15:30 to 18:30 h, where the _{FC} varies from14.6 to 30.58 A. For example, at the time instant equal to 15:30 h, where the _{FC} is 20.35 A, the experimental _{FC} reaches 104 V, whilst the simulated _{FC} is at 108 V. This deviation between both voltage values is equivalent to a relative error of less than 4%. On the contrary, for the time instant equal to 19:00 h, where _{FC} is 20.08 A, the experimental _{FC} reaches 109.5 V, whilst the simulated _{FC} is at 109.2 V. This deviation between both voltage values is equivalent to a relative error of less than 0.5%. Likewise, it can be observed that the simulated _{FC} offers similar dynamics to the experimental _{FC}. _{a} is shown during the test, which gradually increases from 22 to 30 °C during the test. The

This article describes the modelling of a commercial 1.2 kW FC capable of predicting the FC voltage and operating temperature, based on the current demanded and the ambient temperature. The modelling was based on an electrical model and a thermal one. The electrical model proposed is based on the integration of the thermodynamic and activation, ohmic, concentration and double layer phenomena taking place in the FC, likewise the consumption of the peripherals was modelled. Whilst the thermal model proposed is based on the FC thermal energy balance, where the heat generation, dissipation mechanisms and FC thermal capacity were considered. An experimental characterisation was then performed for the electrical and thermal operation alike, making it possible to obtain the parameters for the FC models.

The models were implemented in Matlab Simulink in order to validate these models in a number of steady-state and dynamic operating environments. The validation results are considered to be satisfactory, given the fact that the simulations of the FC models accurately reproduce the electrical and thermal performance of the FCs. Specifically, a mean square error was obtained of less than 0.6 V for the FC voltage prediction, and a mean square error of less than 1.5 °C for the FC operating temperature prediction.

Likewise, the validation of the FC models was performed through the incorporation of four series-connected PEMFC with a rated power of 4.6 kW into a electric microgrid located at the Public University of Navarre. An analysis was made of the operation and the models were validated in a real microgrid operating environment. The results obtained demonstrate, on the one hand, that the FCs have adequate characteristics to adapt to requirements with regard to the power fluctuations caused by consumption and the microgrid generation. And, on the other hand, the model proposed accurately predicts the FC voltage and, therefore, can be used as a simulation tool to carry out the microgrid power output and energy management strategies.

Although, the FCs are able to assume the power fluctuations of the microgrid storage system profile, it may be advisable to incorporate a secondary storage system to provide a rapid response to variations of the _{FC} such as a bank of supercapacitors. In this way the FC would operate in less demanding conditions and, presumably, its useful life would be greater.

We acknowledge the Spanish Ministry of Economy and Competitiveness under grant DPI2010-21671-C02-01 and the Government of Navarre and FEDER funds under project “Microgrids in Navarra: design and implementation”.

The authors declare no conflict of interest.

_{2}and fuel cell buses

_{2}storage needs for early market “man-portable” fuel cell applications

Area (cm^{2})

Parameters associated with activation phenomena

_{i}, b

_{i}

Coefficients associated with activation phenomena

_{dl}

Double layer capacitance (F)

_{p}

Specific heat (J g^{−1} °C^{−1})

_{th}

Thermal capacity (J °C^{−1})

Faraday constant (96485 C mol^{−1})

_{H2}

Hydrogen consumption (NL min^{−1})

_{t}

Heat transfer coefficient (J s^{−1} °C^{−1})

_{v}

Enthalpy of vaporization (J g^{−1})

Current (A)

_{0}

Exchange current (A)

_{DC}

DC current (A)

_{f}

Parameter associated with hydrogen consumption

_{fan}

Parameter associated with fan velocity

_{i}

Parameters associated with peripherals

Mass (g)

Parameters associated with concentration phenomena

_{i}

Coefficients associated with concentration phenomena

Mass flow rate (g s^{−1})

Molar mass (g mol^{−1})

_{s}

Number of series cells

_{fan}

Fan velocity (rpm)

_{fan,min}

Minimum fan velocity (rpm)

Power (W)

Pressure (bar)

_{0},

_{1}

Parameters associated with the hydrogen pressure

Heat flow rate (W)

Resistance (Ω)

_{g}

Ideal gas constant (8.314 J mol^{−1} K^{−1})

_{ohm}

Parameters associated with ohmic phenomena

_{ohm,i}

Coefficients associated to ohmic phenomena

Temperature (°C)

Voltage (V)

_{rev}

Reversible voltage (V)

Impedance (Ω)

Number of electrons transferred in the reaction

^{0}

_{f}

Enthalpy of formation (J mol^{−1})

Transfer coefficient

Alkaline FC

Direct methanol FC

Digital signal processor

Molten carbonate FC

Phosphoric acid FC

Proton exchange membrane FC

Electrochemical impedance spectroscopy

Fuel cell

Frequency response analyzer

Mean absolute percentage error

Power Management System

Root mean square error

Solid oxides FC

Public University of Navarre

Ambient

Activation

Chemical

Concentration

Consumption

Double layer

Evacuated

Fuel cell

Generated

_{2}

Hydrogen

_{2}O

Water

Liquid

Latent

Net

_{2}

Oxygen

Ohmic

Peripheral

Photovoltaic

Reference

Stack

Sensible

Vapour

Wind

(

Electrical model of the FC.

Thermal model of the FC.

Experimental

Results of the individual fit for each temperature and the global fit for the activation parameters (

Small signal model of the electrical performance of the Fuel Cell.

EIS results measured (Meas.) and modelled (Mod.) for a temperature range from 20 to 62 °C.

Results of the individual fit for each temperature and the global fit based on the temperature of the ohmic parameters (

The FC operating temperature evolution for a current range from 10 to 60 A corresponding to a current density range from 0.08 to 0.43 A cm^{−2}.

Fan speed characterisation test: current (_{FC}), FC operating temperature (_{a}), measured fan speed (_{fan} meas.) and simulated speed (_{fan} sim.).

Experimental heat transfer coefficient (_{t} meas.) and fitted (_{t} fit.) for a fan speed range from 900 to 2700 rpm.

Experimental validation of the FC models, drawing a stepped current, with a current variation of 10 to 58 A. (_{FC}) and voltage measured (_{FC} meas.) and simulated (_{FC} sim.); (_{a}).

Dynamic experimental validation test of the FC models: current - voltage of the simulated results (sim) and measured results (meas) for the FC for an operating temperature of 52 °C when required to supply a sine wave current with a peak-to-peak amplitude of 60 A and frequencies of 0.1, 10 and 100 Hz, superimposed on a DC current of 30 A.

Schematic diagram of the microgrid located at the Public University of Navarre.

PV power (_{PV}), wind power (_{W}), power consumed (_{CON}), power of the storage system (_{FC}) measured in the microgrid on the 9th December 2012.

(_{FC}), measured (_{FC} meas.) and simulated (_{FC} sim.) voltage and (_{a}) for the power profile drawn from the microgrid storage system comprising 4 PEMFCs, plotted in

Parameters for the electrical model.

Peripheral consumption | _{0}(A) |
1.5240 |

_{1} |
−1.2080 × 10^{−3} | |

_{2} (A^{−1}) |
4.1180 × 10^{−4} | |

Thermodynamic phenomenon | _{0} (bar) |
1.3240 |

_{1} (bar·A^{−1}) |
−1.3050 × 10^{−4} | |

Activation phenomenon | _{0} (V) |
0.6259 |

_{1} (V·°C^{−1}) |
−1.1128 × 10^{−3} | |

_{0} (V) |
9.1487 × 10^{−2} | |

_{1} (V·°C^{−1}) |
−1.4866 × 10^{−4} | |

Concentration phenomenon | _{0} (V) |
1.8250 × 10^{−2} |

_{1} (V·°C^{−1}) |
−3.3280 × 10^{−5} | |

^{−1}) |
4.500 × 10^{−2} | |

Ohmic phenomenon | _{ohm,0} (Ω) |
2.8959 × 10^{−3} |

_{ohm,1} (Ω·°C^{−1}) |
−4.8479 × 10^{−6} | |

Double layer phenomenon | _{dl} (F) |
4.9183 |

Parameters for the thermal model.

Hydrogen consumption | _{f} (NL·min^{−1}·A^{−1}) |
0.2547 |

| ||

Fan velocity | _{fan,min} (rpm) |
945 |

_{fan} (rpm·°C^{−1}) |
122.9 | |

_{ref} (°C) |
49 | |

| ||

Heat transfer coefficient | _{Ht,1} (W·°C^{−1}·rpm^{−1}) |
0.0136 |

_{Ht,2} (W·°C^{−1}) |
0.6600 |