Based on a review analysis of empirical fuel cell polarization curve models in the literature, an improved model that can predict fuel cell performance with only measured currentvoltage data is developed. The fitting characteristics of this new model are validated by fitting bench test data and road test data. In the case of bench test data, a comparison of the new model and two representative models is conducted, and the results show that the new model presents the best fitting effects over a whole range of current densities. Moreover, the fitted ohmic resistances derived from the new model show good agreement with the measured values obtained through a current interruption test. In the case of using road test data, the new model also presents excellent fitting characteristics and convenience for application. It is the author’s belief that the new model is beneficial for the applicationoriented research of fuel cells due to its prominent features, such as conciseness, flexibility, and high accuracy.
A polymer electrolyte membrane fuel cell (PEMFC) is an electrochemical device that converts chemical energy stored in hydrogen directly to electricity and heat with water as the only byproduct of the reaction. With the prominent features of zero emissions, low operating temperature, quick startup, and high efficiency, PEMFCs have been broadly considered the best electrical energy sources for automotive, stationary, and portable power devices [
For PEMFC, the polarization curve, which describes the relationship between output voltage and current density, is the most important characteristic of performance. Thus, a variety of empirical fuel cell polarization curve models have been developed in the past decades to reproduce the measured polarization curves containing a series of currentvoltage data points. Furthermore, the values of fitting parameters derived from the empirical models are also valuable references for the investigation of PEMFCs.
This work began with a review on existing empirical fuel cell polarization curve models. Based on this, a new empirical model for the entire range of current densities was then presented. Validation of the new model was executed through fitting bench test data and road test data. Regarding bench tests, a polarization curve test and current interruption test were carried out on a fuel cell stack consisting of 57 cells. Taking the measured currentvoltage data as reference values, the fitting accuracy of the new model was investigated and compared with two representative models. Finally, the validity of the new model was discussed by comparing the fitted ohmic resistances with the experimental values. Regarding road tests, some discrete currentvoltage data of a 90cell stack, which were sampled from a demonstrating fuel cell sightseeing vehicle, were used to analyze the application effects of different models.
With the aim of improving fitting accuracy of fuel cell performance throughout the range of operation, a number of researchers have developed numerous empirical and semiempirical fuel cell polarization curve models since the early 1990s. Ten polarization curve models are reviewed in this section. To distinguish fitting parameters from other parameters or constants, fitting parameters are written in boldface throughout this work.
The first empirical fuel cell polarization curve model with five fitting parameters was presented by Kim et al. [
The parameter
Based on model K, several improved models aiming at modifying the term of mass transfer loss have been proposed. The model suggested by Lee et al. [
The model of Squadrito et al. [
The model of Chu et al. [
In this model, first
Similarly, some modifications on the mass transfer loss term were executed by Pisani et al. [
In summary, (
Considering that the models mentioned above cannot accurately fit noload operation and small current densities, Fraser and Hacker [
The reversible voltage, which is a constant term, can be estimated with known fuel cell operation temperature and partial pressures of oxygen and hydrogen. Additional fitting parameters
In the case of a hydrogen/air fuel cell, the thermodynamic reversible voltage,
By substituting the known parameters mentioned above into (
As shown by (
Either (
Equation (
Although there are only four fitting parameters in this model, the limiting current density and current loss are obtained through measurements. As a result, the range of application of this model is limited.
In addition, an analysis technique to evaluate six sources of polarization losses in hydrogen/air proton exchange membrane fuel cells was developed by Williams et al. [
At
The approach in their work is very clear and effective for evaluating different sources of polarization losses under laboratory conditions. Based on determinations of some parameters, including reversible voltage, nonelectrode ohmic resistance, cathode electrode ohmic resistance, limited current density, Tafel slope, and exchange current density, the output voltage of fuel cells is calculated. The development of this model is aimed at distinguishing different polarization losses in fuel cells rather than reproducing the whole polarization curve based on voltage/current data. However, it provided useful insights for analyzing the polarization curve to future works on developing a model without any parameters that lack physical significance.
In conclusion, all of the models reviewed above have their merits and drawbacks, and their numbers of fitting parameters range from 4 to 7. The shortages and inconveniences of these models are listed in Table
Shortages and inconveniences of existing empirical fuel cell polarization curve models.
Features  Equation number  

( 
( 
( 
( 
( 
( 
( 
( 
( 
(  
Number of fitting parameters  5  5  6  5  5  6  6  7  7  4 
Unavailable to fit OCV  √  √  √  √  √  √  
Low fitting accuracy of small current densities  √  √  √  √  √  √  
Requiring estimating 
√  √  √  √  
Requiring measured parameters, such as temperature and gas partial pressure  √  √  √  
Requiring estimating 
√  √  
Mass transfer loss occurring only in high current density region  √  
Requiring other empirical constants  √  
Requiring measured 
√ 
After reviewing existing models of polarization curves, it can be concluded that model K [
In this work, a series of modifications is performed on the simplified version of theoretical fuel cell polarization curve models, and an improved empirical model is then obtained. The processes of modifications are elaborated as follows.
The fuel cell output voltage can be expressed by the following equation:
The activation loss,
Term
The ohmic loss,
The mass transfer loss,
In addition to the activation, ohmic and mass transfer losses, current losses due to hydrogen permeation, and electron crossover also exist in fuel cells. Neglecting the effect of current losses on ohmic loss and mass transfer loss, a simplified version of the theoretical fuel cell polarization curve model that consists of numerous parameters can be described as [
When external current density
The following equation can then result from (
In a fuel cell, as the cell current becomes high, which indicates that the electrochemical reaction rate on the electrode surface is fast, the mass transfer rate of the reactants is not sufficiently fast to provide enough reactants to the electrode surface. Depletion of reactants at the electrode surface leads to a drop in cell voltage. The calculation of the cell voltage drop in this part is difficult in some conditions without all operational parameters, so the empirical equation suggested by Kim et al. [
The term of mass transfer loss,
When a fuel cell is under noload operation, the value of (
Combined with (
Neglecting the mass transfer loss, the new model can be expressed as (
In this simplified formation, there are three fitting parameters:
The new model (denoted model N) can be applied to represent measured polarization curves with only currentvoltage data. Meanwhile, the application of this model does not require reversible voltage and complicated operational parameters, such as the fuel cell temperature and partial pressures of reactants. As a result of the modifications of the logarithmic term and mass transfer loss term, the new model is available to accurately fit very small current densities, and the fitted polarization curve passes through the test OCV absolutely.
The assumptions, approximations, and limitations of the proposed model are illustrated as follows:
Assume that the anode losses of fuel cells are negligible compared with the cathode losses.
Neglect the effect of current losses on ohmic loss and mass transfer loss.
The activation polarization is described by the Tafel equation.
The ohmic loss is expressed by Ohm’s law, and the ohmic resistance remains constant over the whole range of current density.
Assume that the mass transfer loss approximates
The model can be used under all conditions (including different operational conditions and component materials) as long as the voltagecurrent data of polarization curves are measured.
The comparison of the measured and fitted fuel cell performance is carried out with a polarization curve derived from a fresh commercial 8 kW fuel cell stack. The stack consists of 57 cells, which are assembled in series. The active area per single cell is 312 cm^{2}. The fuel cell stack test station for this study is G500, designed by Greenlight Innovation, Canada [
Operational conditions of the fuel cell stack (bench test).
Parameters  Values 

Relative humidity of hydrogen  Without humidification 
Relative humidity of air  80% 
Stoichiometry of hydrogen  1.2 
Stoichiometry of air  2.5 
Hydrogen inlet temperature  60°C 
Air inlet temperature  60°C 
Hydrogen inlet pressure  1.51 atm 
Air inlet pressure  Without pressurization 
Coolant outlet temperature  Approximately 60°C 
The polarization curve test of fuel cell stack was conducted under a constant gas stoichiometry condition. The currentvoltage characteristics were obtained by varying the current density from 0 to the maximum value at a suitable interval and maintaining each current density until the average voltage of a single cell was stabilized within ±5 mV for 3 min [
Measured polarization curve and ohmic resistances of the fuel cell stack.
In addition, the ohmic resistances of the fuel cell stack are obtained through a current interruption test. The average ohmic resistances of single cells, corresponding to different current densities, are acquired by dividing the ohmic resistances of the fuel cell stack by the cell number of 57 and the range from 0.1062 Ω cm^{2} to 0.1086 Ω cm^{2} with an average value of 0.1071 Ω cm^{2}, as depicted in Figure
To validate the new model, the fitting characteristics are investigated and compared with those of model K and model F. Model N and model K can fit the test data directly, whereas model F requires the value of reversible voltage. As shown in Table
The three fitted polarization curves and measured currentvoltage data are depicted in Figure
Comparison of measured and fitted polarization curves (bench test).
Comparison of measured and fitted polarization curves (small current densities) (bench test).
To evaluate the prediction accuracy of these three models in terms of quantity over the whole region of current densities, the fitted voltages of each model are compared with the measured values at different current densities (25 test points in total). For model K, because the fitted voltage corresponding to 0 A/cm^{2} cannot be obtained, the fitted voltage at 1 mA cm^{−2} is used here as a substitution of OCV.
The absolute deviation between fitted voltage and measured voltage is defined as
Absolute deviations of fitted voltage and measured voltage derived from the three models.
The average of the absolute deviations between fitted voltage and measured voltage is defined as (
Average and maximum values of absolute deviations derived from three models.
Model K  Model F  Model N  

Average of absolute deviations (mV)  1.5919  1.4883  1.4642 
Maximum absolute deviation (mV)  3.8224  3.1011  3.4242 
As illustrated in Table
The values of fitting parameters obtained from the three models are listed in Table
Values of fitting parameters obtained from the three models (bench test).
Model K  Model F  Model N  


N/A  1.1940  N/A 

0.7703 ( 
0.9557  0.9560 

0.06362  0.06642  0.06677 

0.1121  0.1083  0.1073 

N/A  0.001225  0.001241 

N/A  3.7980  N/A 

0.005165  0.005248  0.005339 

2.2389  2.2421  2.2353 
During the polarization curve test, currentvoltage data of every single cell in the stack have also been acquired. The polarization curves of 57 single cells are fitted by model N such that the values of fitting parameters can be investigated further.
The ohmic resistances of 57 single cells derived from model N are depicted in Figure
Fitted ohmic resistances of 57 single cells.
Because the ohmic resistance of the fuel cell stack has been obtained through the current interruption test, the average value of the measured results, that is, 0.1071 Ω cm^{2}, is taken as the reference value to the fitted values in what follows. As seen in Figure
Comparison of ohmic resistances obtained from measurement and fits.
From the foregoing analysis, in addition to the excellent fitting characteristics of currentvoltage data, the fitting results of the ohmic resistance derived with model N are in good agreement with the measured values and can be a reference for research on polarization curves.
The model in this work is developed based on a review of other existing models. The fitting accuracy of similar empirical models under different operational conditions and component materials has been corroborated by many other works, such as [
To investigate the performance and durability of fuel cell stacks for automotive applications, road tests or demonstrations of fuel cell vehicles are significant and indispensable. Under a real road environment, a vehicular fuel cell stack experiences much more complicated and harsher operational conditions than those under bench tests, such as dynamic cycling and changes of ambient temperature, pressure, and relative humidity. Because of the frequent load change, the performance data of the fuel cell stack sampled from the road test are massive and discrete, in total contrast to those obtained from the bench test. As a consequence, the empirical fuel cell polarization curve model is an effective tool to extract the whole polarization curve from the complex data.
In this work, some experimental data, which are obtained from a demonstration fuel cell sightseeing vehicle with a sampling frequency of 1 Hz, are used to compare the application effects of the three models, that is, model K, model F, and model N. The fuel cell stack adopted in this sightseeing vehicle has a designed output power of 5.8 kW and consists of 90 cells assembled in series. The electrochemical active area per cell is 250 cm^{2}. Under a real road environment, the known operational conditions of the stack are listed in Table
Operational conditions of the fuel cell stack (road test).
Parameters  Values 

Hydrogen inlet pressure  Approximately 1.6 atm 
Air inlet pressure  Approximately 1.48 atm 
Coolant outlet temperature  Approximately 55°C 
Similarly, model N and model K can fit the discrete data directly, whereas model F requires the value of reversible voltage. According to the information in Table
The raw data depicted in Figure
Comparison of measured and fitted polarization curves (road test).
As plotted in Figure
Comparison of measured and fitted polarization curves (small current densities) (road test).
As seen in Table
Values of fitting parameters obtained from three models (road test).
Model K  Model F  Model N  


N/A  1.2020  N/A 

0.7510 ( 
0.9574  0.9576 

0.06412  0.06532  0.06516 

0.3310  0.3287  0.3292 

N/A  0.0006616  0.0006508 

N/A  1.1926  N/A 
Through the foregoing analysis, it can be concluded that model F and model N show much better fitting characteristics than model K because of the minimum prediction error at OCV and small current density region. Although model N shows almost the same fitting characteristics as model F, the most competitive advantage of model N is that the reactant pressures and operational temperature of the fuel cell stack are not indispensable for fitting. In addition, only three fitting parameters, one less than those of model F, are required for model N. Each fitting parameter in model N has physical meaning, whereas
Among all proposed models, the model in this paper presents good fitting accuracy and the best advantage of convenience collectively. As shown in Table
Inability to fit open circuit voltage.
Low fitting accuracy of small current densities.
Requiring estimate of reversible voltage
Requiring measured parameters, such as temperature and gas partial pressure.
Requiring estimation of the smallest current density that causes the voltage to deviate from linearity
Assuming that mass transfer loss occurs only in the high current density region.
Requiring other empirical constants.
Requiring measured limited current density
The model proposed in this work avoids all shortcomings and inconveniences mentioned above, and it is the author’s belief that this is one of the most meaningful aspects of this work.
The purpose of this work is to obtain a simple, easytouse, efficient, and applicationoriented polarization curve model with an acceptable fitting accuracy based on a review analysis. After summarizing numerous existing empirical fuel cell polarization curve models, a new model has been developed in this paper.
The following conclusions can be obtained:
The new model contains five fitting parameters. Compared with most existing models, it does not require more fitting parameters.
The new model has a compact formation, and its application only requires currentvoltage data that are easy to measure. Hence, it is beneficial for applicationoriented investigations and industrial research.
In the case of using bench test data, the new model provides almost ideal fitting characteristics with the entire range of current densities. Compared with other models, its advantages can be illustrated as follows: (a) the fitting curve definitely crosses the point of measured OCV; (b) the fitting curve presents excellent accuracy over the whole region of current densities. Moreover, all of the values of the fitting parameters derived from the new model appear in reasonable ranges. The fitted ohmic resistances of fuel cells ideally agree with the experimental data obtained through the current interruption test. As a consequence, the fitted results of the parameters can be references for research on fuel cell polarization curves. In addition, at present, under laboratory conditions, it is difficult to precisely measure some parameters, such as the limited current density and internal loss current density. The development of a model without any parameters that lack physical significance is the most important research direction of our future works.
In the case of using road test data, it is difficult to precisely measure the steady pressures of reactants, operational temperature, limited current density, internal loss current density, and so on. As a consequence, the most prominent advantage of the new model is that excellent fitting characteristics over the whole current density region are presented without measuring any pressures or temperatures. Consequently, it is suitable for extracting the fuel cell stack performance accurately and efficiently from the discrete data sampled in a real road test.
The authors declare that there are no competing interests regarding the publication of this paper.
The work was financially supported under the grants of the National Nature Science Foundation of China, Project no. 51275357.