Development of an Equivalent Electronic Circuit Model for PEMFC Voltage Based on Different Input Electrical Currents

The technological development and exploration of the proton exchange membrane fuel cell have relied on several mathematical models. However, despite the wide variety of models, the equivalent electronic circuit model is the most suitable for describing electrical behavior, designing electronic interfaces, and analyzing control and reliability strategies. In addition to the fact that this type of model is scarce to model the fuel cell voltage, an equivalent electronic circuit model that depends only on the input current has not been reported (in general, the reported mathematical models take into account additional variables such as humidity, temperature, pressure, etc.). For this reason, this work focuses on developing an equivalent electronic circuit model for the fuel cell voltage that depends only on the input current. Besides, the configuration of the proposed circuit (one voltage source, two capacitors, and three resistors) is simpler than the circuits proposed in previous works. To validate the model and its parameters, current tests from $1.2~kW$ Nexa® fuel cell power module were used. The comparison between the experimental data and the developed model confirms the efficiency of the equivalent electronic circuit model to reproduce the fuel cell voltage as a function of the current.


I. INTRODUCTION
Concern about greenhouse gas emissions and the problems of fossil fuel consumption has led to the search for renewable energy sources, which aim at being sustainable with affordable prices, highly efficient, and with low environmental impact in energy conversion.Therefore, interest in power generation systems such as Fuel Cell (FC) has increased in recent years.FC systems are clean, environmentally friendly, The associate editor coordinating the review of this manuscript and approving it for publication was Paolo Crippa .and sustainable energy sources due to their higher energy density, energy efficiency, and low emissions [1], [2].
The main operation of the FC is to transform gaseous fuel chemical energy into electricity.FCs can be used as an alternative stationary and mobile power source [3], [4].Today different manufacturers offer many options and types of FC.They can be classified according to their specific characteristics, such as the type of fuel used, the reaction temperature, and the electrochemical material used [5].The main FCs reported in the literature are Proton Exchange Membrane (PEMFC), direct methanol, solid oxide, molten carbonate, phosphoric acid, alkaline, and microbial [6], [7].In the field of both stationary and mobile applications, PEMFCs are among the most widely used and promising.This is due to their many advantages, which are quick and silent start-up operation, robustness, and a relatively low operating temperature ranging from 20 to 80 o C [8].
For the development of PEMFC systems, mathematical modeling is a powerful tool for simulation, exploration of control strategies, prediction, and understanding of power generation system behavior [9], [10], [11].A wide variety of models that can be classified as 0-D to 3-D models [12], empirical and semi-empirical models [13], analytical models [14], statistical physics-based models [15], [16], and performance and fault diagnosis models [17], [18] have been developed.The models that are most suited to the purpose of a microgrid or electrical emulator development are the equivalent Electronic Circuit Models (ECM) since these models allow describing the electrical behavior of the PEMFC or how it interacts with power conditioning circuits such as converters and other connected electrical systems [19].In addition, ECM are a powerful tool for electronic interface design and control as well as for the development of reliability test analysis [20], [21].Two types of ECM have been reported: dynamic models and passive models [19].A dynamic model takes into account both chemical and thermodynamic processes.Thus, it is possible to understand the unique characteristics of the PEMFC (i.e. a clear understanding of various factors including additional losses due to ripple current) to implement the best system in terms of efficiency and cost [22], [23].Furthermore, a dynamic model can be used to optimize the performance, transient response, and efficiency of power converters, so that suitable control systems can be designed to meet the load demand [22], [24].With a passive model, it is possible to determine the performance and degradation of the PEMFC while in standby mode.This mode applies to uninterruptible power systems where the PEMFC is idle most of the time and reliability is paramount [19].
The first ECM has been proposed by Larminie et al. [25].In this first model, the electrodes are each represented by a capacitor in parallel with a resistance and a voltage potential.For the development of ECM, some authors relied on mathematical models that describe the behavior of PEMFC, taking into account several variables that influence the behavior of PEMFC voltage [26], [27], [28], [29].Other authors have chosen to describe the polarization curve taking into account the behavior of the electronic components [22], [24], [30].In [31], the electrochemical behavior of a PEMFC has been taken into consideration for the development of an ECM and a Matlab-Simulink ® model has been established.The authors in [32] used the equivalent double-layer capacitor to avoid step-transient voltage variations.This capacitor has been optimized based on the transient phase convergence and its lifetime.In [33], a state space model of a PEMFC DC/DC converter system has been presented.This model describes the relation between different electrical variables and can be applied to any FC operating point.The authors in [34] presented a simplified and novel dynamic ECM for PEMFC suitable for power system analysis and design.In [35], both the PEMFC and the stacks have been tested under various operating conditions to identify the variations of the model parameters and thus be able to carry out a more detailed analysis of the processes that occur in a PEMFC through the ECM.In [36], an easy-to-use digital simulator for PEMFC based on an ECM using a LabVIEW ® power supply has been presented.The authors in [37] modeled the open-circuit output voltage, voltage losses, mass balance, and thermodynamic energy balance of a PEMFC, along with the formation of the charge double layer at the cathode.In [38], an ECM based on double-layer charge dynamics has been developed assuming a dielectric relaxation of the material to represent dynamic processes in the typical frequency range of the inverter current ripple.In [39], ECM parameters have been estimated based on impedance data sets obtained through electrochemical impedance spectroscopy (EIS) processes.In these processes, a small magnitude alternating current sweeping a wide spectrum of frequencies has been superimposed on a direct current drawn from the PEMFC while the resulting voltage response is measured, which helps to understand the effects of current ripple on the PEMFC while operating with power conditioning units.In [40], an ECM has been developed focused on PEMFC voltage and temperature distribution.The developed model detects, identifies, and simulates different PEMFC failures.In [41], the authors built an ECM based on analogies that allow a uniform representation of electrical, pneumatic, and thermal transients.Furthermore, it has been shown how models representing different transients can be coupled as part of a PEMFC system.In [42], the authors developed an ECM to describe the static-dynamic behavior of the PEMFC stack under pulsed loads up to frequencies of 10 kHz.Thus, with this model, it is possible to perform dynamic simulations of power electronic systems directly connected to the PEMFC stack without the need to parameterize with special EIS measurements.In [43], an electrically controllable ECM for PEMFC is developed and implemented in MATLAB ® .In addition, the authors presented a method to parameterize the models developed in MATLAB-Simulink ® through experimental results.The authors in [44] applied an ECM to fit the EIS measurements of laser coating samples measured at open circuit potential.The diagram of the circuit for this ECM takes into consideration the resistance of the solution, the charge transfer resistance at the electrolyte interface, and the Warburg resistance.In [45], a new diagnostic technique has been developed to determine the hydration of the PEMFC using an ECM.In this case, the electronic circuit proposed by Fouquet et al. [46] was used.The authors in [47] made a review of PEMFC emulators, in which it is shown how the development and construction of emulators for PEMFC voltage is related to ECM.In addition, this work shows examples of electronic circuit designs based on ECM.Table 1 shows the main characteristics of the ECM proposed in recent years (i.e., the variables that have been taken into account for the development of the ECM, the electronic components used in the PEMFC equivalent circuit, and the type of PEMFC used for the development and validation of the ECM).
As can be seen in Table 1, despite the different models proposed, ECM that depends solely on the input current has not been reported.For this reason, this paper makes the following contributions: • Development of an ECM for the PEMFC voltage that depends only on the input current.The proposed ECM holds significant potential in facilitating a comprehensive analysis, designs, optimal sizing, and efficient control strategies for power converters integrated with PEMFC systems.
• An electronic circuit diagram composed of one voltage source, two capacitors, and three resistors, which can be built quickly and cheaply to emulate a PEMFC for prototyping purposes.
To validate the model and its parameters, five different PEMFC current tests of a 1.2 kW Nexa ® PEMFC power module were considered; three increasing and two decreasing currents.Finally, the results of the statistical tests applied to the simulated and experimental data confirmed the efficiency of the ECM to reproduce the PEMFC voltage as a function of the current.Therefore, with the mathematical model developed in this work, it is possible to easily build an electronic circuit for PEMFC voltage emulation.
This work is divided into five sections.After providing the current state-of-the-art and motivations to carry out this work in the Introduction, Section II aims at presenting the PEMFC technology and describes the experiment that was carried out to collect voltage and current data.Then, in Section III, the ECM for PEMFC voltage is provided.In Section IV, the ECM parameters are calculated.In Section V, the ECM is experimentally validated by using data from a 1.2 kW Nexa ® PEMFC power module.Besides, a discussion is provided to summarize and comment on the obtained results.Finally, some concluding remarks are given in Section VI.

II. PEMFC TECHNOLOGY: FEATURES AND MODELING ISSUES A. FEATURES OF PEMFC
A considerable amount of research work has been carried out to develop highly efficient and reliable PEMFCs for use in various applications such as portable and stationary.Significant development has been achieved recently, especially in materials and current density, which will eventually lead to increased power density and make the device more efficient and reliable [48].In addition, PEMFCs have higher efficiency compared to heat engines and in their use in modular electricity generation [3].
A PEMFC consists of a proton exchange membrane sandwiched between two electrodes (anode and cathode).The membrane has a special property that allows protons to pass through while blocking electrons.Hydrogen gas passes over the anode and, with the help of a catalyst, separates into electrons and hydrogen protons.The protons flow to the cathode, through the proton exchange membrane while the electrons flow through an external circuit, thus creating electricity.The hydrogen protons and electrons combine with oxygen flow through the cathode to produce water, see Fig. 1.The reactions in the PEMFC are [37]: Typical characteristics of PEMFC can be identified in a polarization curve, which is a plot of cell current versus cell voltage.As more current is drawn from the FC, the voltage drops due to the electrical resistance of the fuel cell, inefficient transport of reactant gas, and slow reaction rate.Since lower voltage indicates lower fuel cell efficiency, lowload operation is preferred.However, this will increase the volume and weight of the fuel cell.Furthermore, the constant low-load operation is not practical in mobile applications where frequent load changes are required.The polarization curve varies with different operating conditions, including pressure, temperature, partial pressure of reactants, and membrane humidity [3].As is shown in Fig. 1, to form a complete FC system, the PEMFC stack requires several auxiliary components and four main flow subsystems: the hydrogen supply system to the anode; the air supply system to the cathode; the humidifier and the cooler that maintain the humidity degree, and the temperature of the PEMFC.Reactant flow rate, total pressure, reactant partial pressure, membrane temperature, and humidity are the main parameters that need to be regulated to avoid constant warm-ups and ensure fast system transient response, safe shutdown, system robustness, and an adaptation to power changes.The main control devices are the compressor motor for airflow and pressure regulation, the hydrogen flow and pressure regulation valve, the water pump for temperature regulation, and the humidifier for humidity control.It is worth mentioning that changes in one parameter influence the others [3], [49].Thus, this is the cause of a PEMFC being a complex system.

B. EXPERIMENTAL DATA RECORDING
The data for this research was taken from a 1.2 kW Nexa ® PEMFC power module (manufacturer Ballard) [50].This equipment is a fully automated, compact, and lowmaintenance PEMFC stack that provides a power output of 1.2 kW at a voltage of 24 V .Besides, this module incorporates a protection diode to prevent reverse current from entering the FC stack, see Fig.The experiment consisted of observing the behavior of the PEMFC voltage using five different input current tests.The results of this experiment are shown in Fig. 3.
The five current tests were monotonic (tests 1 to 3 consider a current increase from no load to a given maximum current, and tests 4 to 5 consider a current decreasing behavior, from a maximum current to no-load operation), which allows studying the behavior of voltage and power (polarization curve) when the PEMFC is subjected to this type of input current.Notice that the hysteresis band exhibited by the PEMFC voltage presents a complex nonlinear behavior that is difficult to model, therefore it is out of the scope of this work.All tests started with an experimental time of zero.Besides, the three increasing tests had a current initial value of zero while for both decreasing tests the current final value was zero as illustrated in Fig. 4. The main characteristics of each experimental test (i.e.final time, maximum current, minimum voltage, and maximum voltage) are shown in Table 3.After obtaining the experimental data from the PEMFC, a V-I relation was observed.It is noted that when the electric current increases the voltage decreases, and in the same way, when the current decreases the voltage increases.Besides, when developing the model, it has been taken into consideration that the voltage and current in a cell have a maximum (polarization curve).This proposed model only depends on the input current and is explained in detail in the next section.

III. EQUIVALENT ELECTRICAL CIRCUIT AND MATHEMATICAL MODELING
As it is widely known, the PEMFC voltage V fc can be expressed in terms of the Nernst's voltage E th and of the three main types of voltage drops; ohmic V ohm , activation V act , and concentration V con [51], [52].
This equation represents a static model of the PEMFC voltage.This static model has been used by many authors as reported in the literature [53], [54].However, in [40], the authors assumed the PEMFC as an electronic circuit to develop a static-dynamic ECM.This work is inspired by the ECM proposed in [40], wherein the double-layer charging effect serves as the main regulator of the PEMFC voltage dynamics.In this work, the capacitance property was used to model the double layer that corresponds to the electrolyte/electrode contact since it functions as a storage of electrical charges.Also, for the activation and concentration voltages, a voltage change requires a charge time (in case of voltage increase) or a leakage period (in case of voltage decrease).It is important to note that the ohmic voltage drop is not influenced by any time delay as it changes instantly.The equivalent electrical circuit is sketched in Fig. 5.
The equivalent circuit model is composed of the following elements: -Two resistance-capacitor branches to model the dynamics both at the anode and cathode (activation and concentration voltage drops η act + η con ): 108332 VOLUME 11, 2023 Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
where the dynamic equations for η act,c + η con,c and η act,a + η con,a are: where I is the PEMFC current, C c and C a are the capacitors for cathode and anode, respectively.The electrical time constants τ c and τ a governing the dynamics are variables according to the operating conditions at the input of the PEMFC.-One resistance to model the membrane (ohmic voltage drop η ohm ): where R mem is an internal resistance in the circuit.-A DC voltage source V ini to model the Nernst's voltage: The equivalent resistances R c and R a are determined based on the activation voltage and input current of the PEMFC, also they are related with τ c and τ a , respectively, see ( 8) and ( 9) [55], [56], [57].
So, the equivalent static-dynamic model for the PEMFC voltage can be expressed as: 10)

IV. PARAMETER ESTIMATION
In this section, the parameters of ( 10) are estimated.To adequately estimate the parameters, the five different PEMFC current tests (I i , where i takes the value between 1 and 5) were analyzed.Each test I i was estimated using Matlab ® Curve Fitting Application.Among all the functions available in this application, the function with the best fit for each I i was the Fourier series of 4 terms, see (11).
Table 4 shows the different values of the parameters for the Fourier series of each current test I i .
Fig. 6 and 7 illustrate the behavior of the five current tests and the behavior of their corresponding Fourier series.Fig. 6 shows the behavior of the three tests of increasing current   while Fig. 7 shows the behavior of the two tests of decreasing current.
Taking into account the approximation of the electrical current of the PEMFC using the Fourier series, it is possible to calculate the analytical solution of the differential equations (4) and (5).Therefore, for each I i , η act,c,i + η con,c,i and η act,a,i + η con,a,i are represented by the following time functions: where the terms A 0 , A 1 , A 2 , B 0 , B 1 , and B 2 are given by: where η act,c,i,0 +η con,c,i,0 and η act,a,i,0 +η con,a,i,0 are the initial activation and concentration voltages for the cathode and anode for I i , respectively.These voltages are calculated using (20).
where η act,i,0 + η con,i,0 = η act,c,i,0 + η con,c,i,0 + η act,a,i,0 From this result, the parameters of (10) were calculated (i.e. the parameters V ini , R mem , C c , C a , τ c , and τ a ).Due to the monotonic behavior of the polarization curve, it was estimated V ini = max[V fc ] (i.e. the maximum experimental voltage obtained, V ini = V fc (0) for increasing current tests and V ini = V fc (t f ) for decreasing current tests, where t f is the final time of the test).Besides, to facilitate the construction of the electronic circuit, the capacitors must be equal, so it is assumed that C c = C a .
To calculate the parameters R mem and C c , a script was developed using MATLAB ® software.The script and the MATLAB command lsqcurvefit (based on the least-square regression algorithm) were applied to adjust the parameters of the model to the voltage data obtained experimentally for the different electrical current tests.The values obtained were R mem = 0.50781 , and C c = 1.8857F.
According to the references [55], [56], and [57], τ c and τ a are dependent on voltage and current.However, as a result of the script implementation, the values obtained for τ c and τ a were estimated as functions that only depend on the PEMFC voltage as follows.
In the next section, parameters estimation is used to validate the proposed model through a comparison of the numerical simulations with experimental results.

A. SIMULATION AND RESULTS
To assess the effectiveness of the model in reproducing the real behavior of the PEMFC voltage, the relative error E r , the mean error E m , the mean squared error MSE, and the root mean square error RMSE were calculated for all different experimental tests as follows.
where N d is the number of experimental data, V exp,k is the k experimental measurement (V ), and V sim,k is the k simulation data from the model (V ).In Fig. 8    in Fig. 8.In this case, Test 1 had the most significant errors of the statistical tests (i.e., E r = 3.81%, E m = 1.442V , MSE = 1.7264V 2 , and RMSE = 1.3139V ).This result in the statistical tests demonstrated the effectiveness of the model as a function of time in increasing currents.
In Fig. 9, the results obtained through numerical simulations and experimental tests have been compared for the decreasing current tests.In this comparison, Test 5 had the most significant errors of the statistical tests (i.e., E r = 3.15%, E m = 0.9496 V , MSE = 1.2031V 2 , and RMSE = 1.0969V ).This result in the statistical tests demonstrated the effectiveness of the model as a function of time in decreasing currents.
The behavior of the voltage-current model is shown in Fig. 10 and 11.A comparison of numerical simulations of the proposed model with experimental results is reported in Fig. 10.In this comparison, the V-I behavior of the experimental and simulation data can be observed when the input current increases.As can be seen, the PEMFC voltage drops when the PEMFC current increases.
In Fig. 11, a comparison of numerical simulations of the proposed model with experimental results is reported.In this case, the V-I behavior of the experimental and simulation data can be observed when the input current decreases.As can be   seen, the PEMFC voltage increases when the PEMFC current drops.
Finally, the behavior of the error is shown in Fig. 12 and 13, for increasing and decreasing current tests, respectively.As can be seen in both figures, the errors obtained vary between 2.4923 V and 0.050 V .In addition, these errors as a function of time have a random behavior, which increases the difficulty to be modeled analytically.Therefore, applying control theory to minimize the errors obtained is a promising line of research to be addressed as future work.

B. DISCUSSION
As mentioned at the beginning of this section, different tests were carried out to demonstrate the effectiveness of the model.In Tables 5 and 6, the results of the previous subsection are summarized.Table 5 shows the maximum and minimum error values obtained and Table 6 presents the mean, relative mean squared, and root mean square errors.From Table 5, it can be noted that the maximum and minimum errors for the five current tests were for I 1 with a value of 2.4923 V and I 4 with a value of 0.0050 V , respectively.According to Table 6, the mean error, I 1 had the maximum E m with a value of 1.1442 V and I 3 had the minimum E m with a value of 0.6549 V .In the case of relative error, I 2 had the maximum E r with a value of 3.81% and I 3 had the minimum E r with a value of 1.98%.Finally, the tests I 1 and I 3 obtained the maximum and minimum relative mean squared and root mean square errors with values of MSE = 1.7264V 2 , RMSE = 1.3139V and MSE = 0.7292 V 2 , RMSE = 0.8539 V , respectively.It is also worth mentioning that the average value E m and E r of the five current tests were 0.92376 V and 2.984%, respectively.
Therefore, the developed ECM in this work is effective in describing the behavior of PEMFC voltage with different current inputs, both increasing and decreasing currents.Furthermore, using this model and the value of its parameters as a base, it is possible to build an electronic circuit that describes the behavior of Voltage PEMFC and replace the real PEMFC (i.e., construct a PEMFC emulator).However, the estimation of the model parameters is not optimal, and errors between the experimental data and the model can be observed.Besides, the obtained errors are particularly noticeable during the transient phenomena, are complex to model according to current.Thus, taking into account the behavior of the transient phenomenon, it is possible to increase the reliability of the ECM for PEMFC voltage.

VI. CONCLUSION
In this work, the behavior of PEMFC voltage was investigated to develop an accurate dynamic-static ECM.Experimental tests allow identifying the dynamics of the PEMFC voltage for different input currents, both increasing and decreasing.The ECM was developed using five different current tests, three tests of increasing currents and two tests of decreasing currents.To complete the model, the parameters were estimated as constants, except for τ c and τ a , which were estimated as voltage-dependent functions.This new expression for τ c and τ a enriches previous works, where the authors only approximated these parameters as functions dependent on current and voltage.
The precision of the model was validated through different experimental tests, obtaining a mean relative error of 2.984%.The effectiveness of the ECM is illustrated through different experimental tests involving monotonic currents.For future work, testing the effectiveness of the model using currents with non-monotonic behavior would complement this work.
With the ECM developed in this work, different mathematical tools can be applied to improve the performance and optimize the PEMFC, for instance, the design and implementation of a control law.Besides, using the ECM and the calculated parameters, it is possible to build an electronic circuit for PEMFC voltage emulation, thus avoiding the use of a real PEMFC, which brings advantages in various research stages of PEMFC systems.

FIGURE 1 .
FIGURE 1. Schematic diagram of a basic PEMFC.

2 .( 1 )
Unfortunately, this diode exhibits an undesirable power dissipation, resulting in a reduction of effective output power to approximately 1 kW .The output voltage ranges from an open circuit voltage of 42 V to a fully loaded voltage of 22 V .The nominal output voltage is 26 V with a nominal load current of 46 A. The specifications of the studied commercial 1.2 kW Nexa ® PEMFC power module are provided in Table 2.The experimental set-up consisted of the following components: Computer with Matlab ® software and DSPACE DS1103 data acquisition system.(2) Digital oscilloscope for monitoring.(3) Electronic load.(4) Current amplifier probe.(5) A 1.2 kW Nexa ® PEMFC power module (6) Voltage source, LV25P voltage sensor, CSNE151 current sensor.(7) Op-amp signal conditioning circuit.

FIGURE 6 .
FIGURE 6. Behavior of Fourier series and current tests I 1 , I 2 , and I 3 .

FIGURE 7 .
FIGURE 7. Behavior of Fourier series and current tests I 4 and I 5 .
and 9, a comparison of numerical simulations of the proposed model with experimental results is shown.Increasing current tests were used for the comparison shown

FIGURE 8 .
FIGURE 8. Behavior and comparison of the model with the experimental data for current tests I 1 , I 2 , and I 3 .

FIGURE 9 .
FIGURE 9. Behavior and comparison of the model with the experimental data for current tests I 4 and I 5 .

FIGURE 10 .
FIGURE 10.Behavior of the voltage-current model for current tests I 1 , I 2 , and I 3 .

FIGURE 11 .
FIGURE 11.Behavior of the voltage-current model for current tests I 4 and I 5 .

FIGURE 12 .
FIGURE 12. Behavior of the error obtained for current tests I 1 , I 2 , and I 3 .

FIGURE 13 .
FIGURE 13.Behavior of the error obtained for current tests I 4 and I 5 .

TABLE 1 .
Main characteristics of the ECM proposed in recent years.

TABLE 2 .
[50]ifications of 1.2 kW Nexa ® PEMFC power module (manufacturer Ballard)[50].FIGURE 3. PEMFC voltage behavior with five different input current tests.Behavior of PEMFC current tests as a function of time.

TABLE 3 .
Main characteristics of the five experimental tests.

TABLE 4 .
Fourier series parameters for each current test.

TABLE 5 .
Values of the maximum and minimum errors for each test.

TABLE 6 .
Results the statistical tests applied.