A Real Case: How to Combine Polarization Curve and EIS Techniques to Identify Problematic Cells in a Commercial PEM Stack

Abstract

Nowadays, the mobility sector is assessing different technologies to substitute the internal combustion engines in order to reduce its CO₂ emissions; one of these possible alternatives is the Polymer Electrolyte Membrane (PEM) fuel cell. So, the development of non-destructive diagnostic tools that could identify defective cells and/or any malfunctioning behavior and can be easily embarked on in any vehicle will expand the durability of PEM fuel cells, improve their performance, and enable them to carry out predictive maintenance. In this research, we use an in-house developed methodology that combines the polarization curve and electrochemical impedance spectroscopy (EIS) techniques to characterize different cells of a commercial PEM stack, identifying malfunctioning ones.

1. Introduction

Currently, the global mobility sector is sifting to electrification in order to mitigate the environmental effects partially caused by internal combustion engine vehicles, such as global warming and/or air pollution [1,2,3]. In this new context, different electric vehicle technologies have appeared, which can basically be classified into five categories: battery electric vehicles (BEVs), plug-in hybrid electric vehicles (PHEVs), hybrid vehicles (HVs), fuel cell electric vehicles (FCEVs), and extended-range electric vehicles (ER-EVs) [4], but only the BEVs and FCEVs are fully electric, as the other technologies combine electric and conventional combustible engines.

So, the fuel cells together with batteries are called to play a leading role to address the decarbonization of this sector [5], especially if hydrogen is produced from renewable energies [6,7,8,9]. And among the fuel cell technologies, the PEM fuel cells are the most suitable for mobility applications [10,11,12] because of their characteristics, operating temperature, easy scale-up, all solid components, fast dynamic response, high energy conversion efficiency, and power density.

Therefore, [13,14,15] development of non-intrusive and easily embarked diagnostic tools that could assess the performance of stack cells during real operation, identifying anomalous behavior, will ease the transition from combustion engines to this more friendly environmental technology, as these tools will enable to carry out predictive maintenance, expanding the durability of PEM fuel cells, and improving their performance.

According to [16], fuel cell failure diagnosis tools can be classified into two categories: off-line and on-line techniques.

The off-line category refers to those methodologies in which the analyzed fuel cell is operated under some conditions that do not necessarily match with the normal operating fuel cell conditions. This category includes electrochemical characterization techniques such as cyclic voltage (CV), galvanostatic charge (GSC), electrochemical impedance spectroscopy (EIS), etc.

On the other hand, the on-line diagnosis methodologies are able to infer the fuel cell state of health when it is operated in their intended application. In Refs. [13,16,17] these techniques use some mathematical algorithms to process in real time different sensor readings into a diagnosis. According to the used algorithms, those techniques can be classified into: model-based, data-driven, signal-based, and hybrid methods. Where the model-based methods consist in a differential analysis between real-time measurements and healthy fuel cell model outputs.

In this research, we use an in-house developed methodology, which is described in [18], to detect malfunctioning cells of a commercial stack. This method is based on a five-parameter equation, which is able to model the voltage of a cell in function of current. And, using the results of EIS and polarization curve tests to determine these five parameters for this cell, we transpose the test conditions (temperature, pressure, relative humidity, etc.) to the model and obtain for this cell its open circuit voltage (OCV), internal resistance, and an expression for its activation losses. And therefore, comparing these results of a cell with the results for a healthy one, we can try to determine if the assessed cell shows flaws.

According to the previous paragraphs, our methodology can be classified as model-based, but on the other hand, it cannot be considered on-line because the used EIS technique is sensitive to electric noise, so it should not be performed during transient events, and the normal driving activity is plenty of them. Even though the EIS technique used in this research is an off-line one, currently there are some efforts to develop on-line diagnosis tools based on the EIS technique [19] that may be used to implement the fuel cell analysis methodology described in this article.

The choice of the selected techniques to characterize a cell was based on the fact that the necessary equipment to perform tests should be easily embarked upon and not complex, and both techniques fulfill this requirement, and additionally, they are common fuel cell electrochemical characterization techniques and used to diagnostic (PEM) fuel cells and stacks [20,21,22,23,24,25,26].

The polarization curve test [27,28,29] probably is the most basic PEM fuel cell characterization test. In this test, the voltage of the fuel cell is measured after stationary behavior is reached, when the fuel cell is forced to work at specific current and test conditions (temperature, pressure, gas relative humidity, gas stoichiometry, etc.). Synthesizing the general performance of the fuel cell into a voltage signal can be used to determine the fuel cell state of health, but on the other hand, this integration process can blur the causes of the fuel cell behavior.

Electrochemistry Impedance Spectroscopy [30,31,32] is a technique that tries to cause resonance with subprocesses of an electrochemical system and study its response, obtaining extra-information that is not accessible for non-resonance techniques such as polarization curve tests. To this end, an electrical stimulus (a known tiny periodic voltage or current signal) is applied to electrodes at different frequencies, obtaining a response (a resulting current or voltage). And this extra information makes the EIS technique suitable to detect some internal flaws as hydrogen crossover flow, study the effects of operating conditions on the fuel cell internal resistance, and determine the hydric conditions of fuel cells [33,34,35,36].

The aim of this research is to test the sensitivity of the methodology described in [18] to identify flaws in a stack. To this end, we used a commercial PEM stack, in which we previously detected a gas leakage in the seal of one cell when the stack temperature dropped below 15 °C. Then, fixing the ambient temperature at 20 °C and applying our method to the cell with flaws and on the other two normal cells, and comparing results, we observed differences between the internal resistance and the expressions for activation losses of the unhealthy and healthy cells.

2. Materials and Methods

In this section, we will briefly describe the mathematical expression used to model the cell behavior, the polarization curve and EIS test procedure, and the test setup with a description of the main equipment.

2.1. Mathematical Model Description

To model the cell behavior, we use Equation (1)

$$V=V_0-Cte{\displaystyle \frac{I}{I^b+d}}-r·I$$

(1)

where V is the cell voltage; I is the current; and $V_0$, $Cte$, $b$, $d$ and $r$ are the fitting parameters.

To obtain $b$ and $d,$ we use the equivalent circuit of Figure 1 to fit the EIS results, where $R_{\Omega }$ is the ohmic resistance, $R_{tc}$ is charge transfer resistance, and CPE = constant phase element. Now, we plot $R_{tc}$ as function of current and fit the resulting curve by Equation (2), obtaining b and d. $\mathcal{V}$ is a fitting parameter of Equation (2).

$$R_{tc}={\displaystyle \frac{\mathcal{V}}{I^b+d}}$$

(2)

Then, substituting $b$ and $d$ in Equation (1) and using this expression to fit the polarization curve, we achieve the rest of parameters, obtaining the following correspondences $V_0=$ open circuit voltage (OCV), $r=$ internal resistance, and ${\displaystyle \frac{Cte}{I^b+d}}$ mathematical expression for activation losses.

It is convenient to highlight that whenever we have used this methodology, the fitting parameters of Equation (1) have always had physical meaning if previously we have obtained $b$ and $d$ through Equation (2). Conversely, if we use Equation (1) without restrictions, the fitting is slightly better, but in this case, the fitting parameters have no physical sense.

A more comprehensive explanation of the model and the development of the mathematical expressions are included in [18].

2.2. Test Setup and Main Equipment Description

In this research, we tested an used commercial fuel cell system of 2500 W, which has a stack composed of 80 cells connected in series. It is an autonomous system that, for operation, only requires an external hydrogen supply, and it is designed to power applications such as aerial drones, ground robots, portable power packs, etc.

We selected this used unit because a tiny H₂ leakage was observed in the vicinity of Cell 79 when the stack was tempered below 15 °C before starting up, providing us the opportunity to check if the methodology described in [18] was able to detect any anomalous behavior in Cell 79. Then, to this end, we applied this methodology to Cell 2, Cell 40, and Cell 79 with the aim of performing a differential analysis.

In Figure 2, there is a scheme of the test setup used to perform the tests. It basically consisted of a H₂ supply system, an electronic load to simulate the power demand, a power supply to feed the Fuel Cell System BOP, a control station to command the Fuel Cell System and the electronic load, and EIS equipment. This EIS equipment was composed of the following:

• Potentiostat: Solartron SI1287 (Electrochemical Interface).
• Frequency analyzer: Solartron SI1255 HF (Frequency Response Analyzer).

Figure 2. Test setup.

Figure 2. Test setup.

To analyze the EIS results, we used the Zplot and Zview software version 3.5h by Scribner.

These tests basically consisted of forcing the fuel cell system to work at some currents, and after fuel cell voltage stabilization, we launched a frequency profile, obtaining the impedance response for each of the selected cells. To perform these tests, the stack was tempered at 20 °C.

3. Results

In this section, we analyze the results of applying the methodology described in [18] on Cell 2, Cell 40, and Cell 79.

3.1. Test Results for Cell 79

Figure 3 plots the polarization and EIS curves obtained for Cell 79.

To better appreciate the EIS curves, we represent them individually and their fitting curve in Figure 4. To fit the EIS curves, we use the equivalent circuit of Figure 1.

If we analyze Figure 4, we can observe the existence of anomalous points. To discriminate these points, we calculate the distance between the measured and fitted values for each point through Equation (3).

$$dis=\sqrt{{\left({\displaystyle \frac{Z_{meas}^{\prime }\left(i\right)-Z_{fitt}^{\prime }\left(i\right)}{\max \left(Z_{meas}^{\prime }\right)-\min \left(Z_{meas}^{\prime }\right)}}\right)}^2+{\left({\displaystyle \frac{Z_{meas}^{″}\left(i\right)-Z_{fitt}^{″}\left(i\right)}{\max \left(Z_{meas}^{″}\right)-\min \left(Z_{meas}^{″}\right)}}\right)}^2}$$

(3)

where $Z_{meas}^{\prime }\left(i\right)$ is the measured impedance real part of i-point; $Z_{meas}^{″}\left(i\right)$ is the measured impedance imaginary part of i-point; $Z_{fitt}^{\prime }\left(i\right)$ is the fitted impedance real part of i-point; $Z_{fitt}^{″}\left(i\right)$ is the fitted impedance imaginary part of i-point; $\max \left(Z_{meas}^{\prime }\right)$ is the maximum value of all measured impedances real parts; $\min \left(Z_{meas}^{\prime }\right)$ is the minimum value of all measured impedances real parts; $\max \left(Z_{meas}^{″}\right)$ is the maximum value of all measured impedances imaginary parts; $\min \left(Z_{meas}^{″}\right)$ is the minimum value of all measured impedances imaginary parts.

Then, define the outlier as the point that is 1.25 from the mean distance in Equation (4), where N is the total number of points for an EIS curve.

$$dis>1.25·{\displaystyle \frac{1}{N}}\sum _{i=1}^{N}dis$$

(4)

And suppressing the points that accomplish this condition, we obtain Figure 5. This criterion is not applied to the EIS curve at I = 0 A.

Now, we use the equivalent circuit of Figure 1 to fit the EIS curves of Figure 5, obtaining the following parameterization,

Table 1. Fitting parameter values for the equivalent circuit of Figure 1, Cell 79. T (F) and P are typical parameters of a CPE impedance.

I (A)	RΩ (Ω)	Rtc (Ω)	T (F)	P
0.00 ± 0.01	0.106 ± 0.001	0.854 ± 0.010	3.134 ± 0.032	0.854 ± 0.010
0.50 ± 0.01	0.107 ± 0.000	0.102 ± 0.002	2.611 ± 0.063	0.816 ± 0.010
1.00 ± 0.01	0.107 ± 0.000	(5.309 ± 0.070)·10−2	2.279 ± 0.068	0.833 ± 0.010
2.00 ± 0.01	0.107 ± 0.000	(2.907 ± 0.036)·10−2	1.971 ± 0.075	0.854 ± 0.011
5.00 ± 0.01	0.107 ± 0.000	(1.414 ± 0.002)·10−2	1.600 ± 0.078	0.898 ± 0.013
10.00 ± 0.01	0.106 ± 0.000	(8.679 ± 0.201)·10−3	1.396 ± 0.135	0.945 ± 0.023
20.00 ± 0.01	0.106 ± 0.000	(6.478 ± 0.212)·10−3	1.287 ± 0.186	0.955 ± 0.032
30.00 ± 0.01	0.106 ± 0.000	(5.915 ± 0.312)·10−3	1.961 ± 0.438	0.911 ± 0.052
40.00 ± 0.01	0.106 ± 0.000	(6.721 ± 0.352)·10−3	3.028 ± 0.609	0.839 ± 0.050
50.00 ± 0.01	0.106 ± 0.000	(7.401 ± 0.424)·10−3	3.117 ± 0.700	0.863 ± 0.056

.

Now we represent R_tc against current and fit the resulting curve by Equation (2), obtaining the following:

$$R_{tc}={\displaystyle \frac{0.06034}{I^{0.9084}+0.07066}}\hspace{1em}〈er{r}_{rel}〉=27\%$$

(5)

Defining the mean relative error by Equation (6)

$$<{err}_{rel}>={\displaystyle \frac{1}{N}}\sum _{i=1}^N{\displaystyle \frac{\left|x_{i\_meass}-x_{i\_fitt}\right|}{x_{i\_meass}}}$$

(6)

where $x_{i\_meass}$ are measured values, $x_{i\_fitt}$ are fitting values, and N is the number of measurements.

In Figure 6, we represent the $R_{tc}$ versus current and its fitting curve; the left picture corresponds to the whole current range, while the right picture only shows the curves from I = 0.5 A to I = 50 A to better appreciate the discrepancies between measurement and fitting curve, observing that Equation (5) only describes correctly the measurements for low currents below 10 A.

Then, substituting $b=0.9084$ and $d=0.07066$ in Equation (1) and fitting the polarization curve of Figure 3, we obtain Equation (7):

$$V=0.9585-{\displaystyle \frac{0.13133}{I^{0.9084}+0.07066}}·I-0.00253·I ; 〈e{rr}_{rel}〉=0.009$$

(7)

Now, in Figure 7, we plot together the polarization curve of Figure 3 and its fitting curve and observe that they are practically coincident, with a mean relative error lower than 1%.

Therefore, we have the following characteristic parameterization for Cell 79:

• Open circuit voltage ${(V}_0)$: 0.9585 V.
• Internal ohmic resistance $\left(r\right)$: 0.00253 Ω.
• Mathematical expression of activation voltage loss: ${\displaystyle \frac{0.13133}{I^{0.9084}+0.07066}}$.

3.2. Test Results for Cell 40

In Figure 8, we represent the polarization and EIS curves obtained for Cell 40.

As to Cell 2, we represent individually the EIS and their fitting curves in Figure 9. To this end, we use the equivalent circuit of Figure 1.

Now, using Equations (3) and (4) to suppress the anomalous points from the analysis and fitting the resulting curves by the equivalent circuit of Figure 1, we obtained the following parameters for the filtered EIS curves,

Table 2. Fitting parameter values for the equivalent circuit of Figure 1, Cell 40. T (F) and P are typical parameters of a CPE impedance.

I (A)	RΩ (Ω)	Rtc (Ω)	T (F)	P
0.00 ± 0.01	(9.37 ± 0.06)·10−2	1.163 ± 0.018	2.973 ± 0.036	0.844 ± 0.007
0.50 ± 0.01	(9.44 ± 0.03)·10−2	0.104 ± 0.002	2.336 ± 0.068	0.834 ± 0.012
1.00 ± 0.01	(9.35 ± 0.02)·10−2	(5.971 ± 0.119)·10−2	2.327 ± 0.086	0.815 ± 0.012
2.00 ± 0.01	(9.20 ± 0.02)·10−2	(3.278 ± 0.044)·10−2	1.875 ± 0.076	0.856 ± 0.012
5.00 ± 0.01	(9.00 ± 0.01)·10−2	(1.586 ± 0.020)·10−2	1.521 ± 0.073	0.891 ± 0.012
10.00 ± 0.01	(8.91 ± 0.02)·10−2	(9.480 ± 0.218)·10−3	1.192 ± 0.109	0.948 ± 0.022
20.00 ± 0.01	(8.81 ± 0.01)·10−2	(6.478 ± 0.212)·10−3	1.287 ± 0.186	0.955 ± 0.032
30.00 ± 0.01	(8.80 ± 0.01)·10−2	(5.738 ± 0.233)·10−3	1.512 ± 0.270	0.942 ± 0.040
40.00 ± 0.01	(8.76 ± 0.02)·10−2	(5.950 ± 0.321)·10−3	1.760 ± 0.420	0.928 ± 0.054
50.00 ± 0.01	(8.73 ± 0.03)·10−2	(6.310 ± 0.363)·10−3	2.408 ± 0.550	0.861 ± 0.054

.

Now we represent R_tc against current and fit the resulting curve by Equation (2), obtaining

$$R_{tc}={\displaystyle \frac{0.06365}{I^{0.83056}+0.05473}}\hspace{1em}〈er{r}_{rel}〉=18\%$$

(8)

And proceeding as Section 3.2, we fit the polarization curve of Figure 8, obtaining

$$V=0.96483-{\displaystyle \frac{0.14259}{I^{0.83056}+0.05473}}·I-0.00118·I ; 〈e{rr}_{rel}〉=0.004$$

(9)

Then, if we plot together the polarization and the fitting curves in Figure 10, we can observe they can be considered overlapped:

Therefore, the characteristic parameters of Cell 40 are as follows:

• Open circuit voltage ${(V}_0)$: 0.96483 V.
• Internal ohmic resistance $\left(r\right)$: 0.00118 Ω.
• Mathematical expression of activation voltage loss: ${\displaystyle \frac{0.14259}{I^{0.83056}+0.05473}}$.

3.3. Test Result for Cell 2

Figure 11 shows the polarization and EIS curves for Cell 2.

Now, as usual, we represent individually the EIS curves and their fitting curves in Figure 12. To fit the EIS curves, we use the equivalent circuit of Figure 1.

As before, we remove the anomalous points from the analysis through Equations (3) and (4), and using the equivalent circuit of Figure 1, we fit the filtered EIS curves, obtaining the parameters of

Table 3. Fitting parameter values for the equivalent circuit of Figure 1, Cell 2. T (F) and P are typical parameters of a CPE impedance.

I (A)	RΩ (Ω)	Rtc (Ω)	T (F)	P
0.00 ± 0.01	(9.37 ± 0.06)·10−2	1.163 ± 0.018	2.973 ± 0.036	0.844 ± 0.007
0.50 ± 0.01	(9.90 ± 0.03)·10−2	0.107 ± 0.002	2.337 ± 0.074	0.829 ± 0.013
1.00 ± 0.01	(9.80 ± 0.03)·10−2	(5.969 ± 0.103)·10−2	2.231 ± 0.080	0.826 ± 0.012
2.00 ± 0.01	(9.65 ± 0.02)·10−2	(3.236 ± 0.043)·10−2	1.895 ± 0.078	0.850 ± 0.012
5.00 ± 0.01	(9.43 ± 0.02)·10−2	(1.514 ± 0.022)·10−2	1.460 ± 0.081	0.904 ± 0.015
10.00 ± 0.01	(9.36 ± 0.01)·10−2	(9.246 ± 0.181)·10−3	1.225 ± 0.101	0.950 ± 0.020
20.00 ± 0.01	(9.29 ± 0.01)·10−2	(6.447 ± 0.186)·10−3	1.261 ± 0.160	0.960 ± 0.028
30.00 ± 0.01	(9.27 ± 0.01)·10−2	(5.700 ± 0.265)·10−3	1.559 ± 0.325	0.944 ± 0.046
40.00 ± 0.01	(9.21 ± 0.02)·10−2	(5.431 ± 0.299)·10−3	1.561 ± 0.389	0.963 ± 0.056
50.00 ± 0.01	(9.18 ± 0.05)·10−2	(7.266 ± 0.732)·10−3	4.209 ± 1.5031	0.779 ± 0.091

.

As to the other cells, we represent R_tc against current and fit this curve by Equation (2), resulting in

$$R_{tc}={\displaystyle \frac{0.06416}{I^{0.86177}+0.05516}}\hspace{1em}〈er{r}_{rel}〉=20\%$$

(10)

Now proceeding as usual, we fit the polarization curve of Figure 11, obtaining

$$V=0.9631-{\displaystyle \frac{0.14453}{I^{0.86177}+0.05516}}·I-0.00165·I ; 〈e{rr}_{rel}〉=0.006$$

(11)

In Figure 13, we plot together the polarization curve of Figure 11 with its fitting curve, obtaining a mean relative error between curves of 0.006.

Therefore, according to the obtained parametrization, Cell 2 has the following characteristics:

• Open circuit voltage ${(V}_0)$: 0.9631 V
• Internal ohmic resistance $\left(r\right)$: 0.00165 Ω
• Mathematical expression of activation voltage loss: ${\displaystyle \frac{0.14453}{I^{0.86177}+0.05516}}$

4. Discussion

In Figure 14, we plot together the polarization curves of Cell 2, 40, and 79.

If we analyze Figure 14, we can observe that they show similar behavior; their differences are slightly with a relative mean standard deviation <2%. So, we can conclude the polarization curve test is not able to detect any flaws in Cell 79; even Cell 70 shows the best performance.

On the other hand, if we compare the values of open circuit voltage (OCV), internal resistance, and activation loss expression for each cell, we can observe discrepancies between cell behavior.

Before continuing with discussion, it is convenient to notice that when we use Equation (1) to fit the polarization curves, the fitting parameters $V_0$, $Cte$, and $r$ are completely free, and we have not imposed any restrictions on them.

4.1. Open Circuit Voltage ${(V}_0)$ Discussion

To analyze the V₀, we plot together the measured and fitted open circuit voltages for each cell, obtaining Figure 15:

Then, it is easy to observe that the measured and fitted V₀ can be considered coincident, so the methodology described in [18] provides very good results for open circuit voltage.

4.2. Internal Ohmic Resistance $\left(r\right)$ Discussion

In this case we do not have a direct measure of the cell internal ohmic resistance; the R_Ω (Ω) of the EIS test includes the effect of the contact resistance between electrodes and stack. Then, taking into consideration that the internal ohmic resistance is predominant in the linear part of the polarization curve, we use the next mathematical expression to calculate an estimation of it for each cell.

$$r_{estimated}={\displaystyle \frac{V_{10\mathrm{A}}-V_{50\mathrm{A}}}{50 \mathrm{A}-10 \mathrm{A}}}$$

(12)

where $V_{10\mathrm{A}}$ is the cell voltage at 10 A and $V_{50\mathrm{A}}$ is the cell voltage at 50 A. Now in Figure 16, we plot the estimated and fitted internal ohmic resistance for each cell.

In this case we can observe the estimated and fitted internal ohmic resistances are not coincident, but they have the same magnitude order, and the estimated resistances are always higher than the fitted ones, as it would be expected because in the estimated resistance is included the effect of the activation voltage loss, while in the fitted ones this effect has been suppressed.

But the most interesting results come out of the fitting internal resistances in comparison. We can observe that the lower value corresponds to Cell 40, and it makes sense as Cell 40 is internal, and it would be expected that the contact between internal cells was more intimate than external ones. But the internal resistance of Cell 79 is excessive; it multiplies by 1.53 the internal resistance of Cell 2 and by 2.14 the internal resistance of Cell 40, and this fact is consistent with a H₂ leakage in Cell 79; bigger resistances could imply the existence of not correct tightened cells, and this could cause H₂ leakage.

4.3. Activation Voltage Loss $\left(AVL\right)$ Discussion

According to dimensional analysis, the kinetic voltage loss is an ohmic resistance, and it should be related to the transfer charge resistance ($R_{tc}$). To check this relation, and taking into consideration that the effect of the AVL on the cell voltage is more determinant for low currents, we compare the AVL and $R_{tc}$ values for the [0–5] A range,

Table 4. R_tc (Ω) and AVL (Ω) values of Cell 2, Cell 40, and Cell 79 for the [0–5] current range.

Rtc (Ω)				AVL (Ω)
	Cell 2	Cell 40	Cell 79		Cell 2	Cell 40	Cell 79
0.0 A	1.1630	1.1630	0.8540	0.0 A	2.6202	2.6053	1.8586
0.5 A	0.1070	0.1040	0.1020	0.5 A	0.2387	0.2311	0.2176
1.0 A	0.0597	0.0597	0.0531	1.0 A	0.1370	0.1352	0.1227
2.0 A	0.0324	0.0328	0.0291	2.0 A	0.0772	0.0778	0.0674
5.0 A	0.0151	0.0159	0.0141	5.0 A	0.0356	0.0369	0.0299

and Figure 17.

Now, comparing the AVL and $R_{tc}$ values, we observe they are different, but they show similar. Therefore, the AVL mathematical expression reproduces the behavior of transfer charge resistance.

But the most interesting result is that the behavior of the AVL for Cell 79 is clearly different to Cell 40 and Cell 2, while the AVL for Cell 40 and Cell 2 can be considered coincident. Cell 79 shows the best performance as it shows the lowest resistance, or, in other words, Cell 79 is more activated than Cell 2 and Cell 40. Then, if we calculate the AVL value of all cells when I = 0 A, we obtain

$${AVL}_{cell2\_I=0\mathrm{A}}=2.62 \Omega ; {AVL}_{cell40\_I=0\mathrm{A}}=2.61 \Omega ; {AVL}_{cell79\_I=0\mathrm{A}}=1.86 \Omega$$

Now, taking into consideration that the AVL value decreases with current, we can calculate at which current the AVL value of Cells 2 and 40 is equal to $1.86 \Omega$.

$${AVL}_{cell2}={\displaystyle \frac{0.14453}{I^{0.86177}+0.05516}}=1.86\to I=0.012 \mathrm{A}$$

$${AVL}_{cell40}={\displaystyle \frac{0.14259}{I^{0.83056}+0.05473}}=1.86\to \mathrm{I}=0.010 \mathrm{A}$$

Then, according to these results and our interpretation, the standby state of Cell 79 corresponds to the state of Cell 2 and Cell 40 when they are producing a current of I = 0.012 A and I = 0.010 A, respectively. This may be interpreted as the Cell 79 is more activated than the rest of the cells to compensate for the voltage loss caused by the internal resistance, and this extra work probably accelerates the degradation processes of Cell 79. If this interpretation is correct, this methodology may be a useful tool for predictive maintenance, but due to the current stage of our research, we are not in a position to forecast the durability of any defective cell. To this end, we would need to study how the model parameters evolve according to how the stack is used for its intended purpose, but this task will be faced in future research.

In light of these results, the usefulness of this methodology could be extended to other topics of fuel cell systems, such as the implementation of intelligent control logics in this kind of system. According to [37], a fuel cell system is a compound of several subsystems, and implementing an intelligent control would be desirable to obtain an accurate and reliable fuel cell model that could be able to provide the static and dynamic response of a FC in real operation. Currently, to obtain this model, there would be two approaches: physical and empirical models. The physical models, which try to model the physical phenomena of a fuel cell, are complex and require large computational resources; on the other hand, the empirical models are simpler, although they can provide accurate behaviors, their parameters could not be interpreted from a physical point of view.

In this research, we have developed an empirical and simple mathematical fuel cell model whose parameters have physical meaning. But currently our methodology is not able to reproduce the dynamic behavior of a fuel cell, as the used EIS technique does not provide correct parameterization in non-stationary tests. So the development of EIS techniques that could be used during dynamic testing could make our methodology a promising candidate for fuel cell control models.

Additionally, if we analyze the fuel cell model implemented through Equation (1), we find that probably its most remarkable characteristic is that the fitting parameters have physical meaning, so they could be useful to obtain an insight into a fuel cell internal phenomena. Conforming to [38] and [39], one of the biggest concerns with PEM Fuel Cell is improving the kinetics of cathode catalysts because the reaction kinetics of ORR are much slower than the HOR. So currently there are some efforts to obtain high-performance electrocatalysts for ORR with a reduced quantity of Pt. In this context, our research could play a role, as the AVL term of Equation (1) may be useful to define a metric to quantify the performance for different ORR electrocatalyst solutions.

5. Conclusions

Then, summarizing the most remarkable results of this research, we conclude that:

According to this research, the polarization curve test is not able to detect this type of flaw, as Cell 79 shows the best performance (Figure 14). But on the other hand, the methodology described in [18] is sensitive, the internal ohmic resistance of Cell 79 is clearly bigger, and the behavior described by AVL expression of this cell is different from the others. Therefore, this methodology could be a useful off-line tool for finding incorrectly tightened cells and/or detecting over-activated ones.

Conforming to the results, this technique provides excellent values for OCV and at least acceptable values for r and the AVL expression reproduces correctly the behavior of $R_{tc}$ for low currents below 5 A.

Equation (1) consistently provides good approximations for the polarization curves, but if the parameters b and d take the values calculated using the EIS results, the fitting parameters of Equation (1) have always physical sense. According to the article, the following are true:

• $V_0$: Open circuit voltage.
• $r$: Internal ohmic resistance.
• ${\displaystyle \frac{Cte}{I^b+d}}$: Mathematical expression for activation voltage loss (AVL).

The physical meaning of AVL parameters is more controversial. The Cte parameter can be thought of as a product of a proportional constant (PC) by $\mathcal{V}$, or, in other words

$$AVL={\displaystyle \frac{Cte}{I^b+d}}=PC·R_{tc}=PC·{\displaystyle \frac{\mathcal{V}}{I^b+d}}$$

PC is a proportional constant that provides the effective loss of voltage caused by $R_{tc}$. The rest of parameters $\mathcal{V}$, b, and d are used to model the $R_{tc}$ dependence with current. According to our interpretation of the fuel cell model represented in Figure 1, the AVL is modeled by the CPE and $R_{tc}$, and when the fuel cell is operated at OCV, only a tiny residual current would cross the cathode double layer (CPE element), but as the power grows, more current would cross the double layer in parallel pathways, as it is represented in Figure 18.

Then the total charge transfer resistance can be interpreted as the sum of parallel resistances of the ionic current paths that cross the double layer, whose number increases with current:

$${\displaystyle \frac{1}{R_T}}={\displaystyle \frac{1}{R_1}}+{\displaystyle \frac{1}{R_2}}+\dots +{\displaystyle \frac{1}{R_n}}\to {\displaystyle \frac{1}{R_T}}={\displaystyle \frac{R_2·R_3···R_n+R_1·R_3···R_n+\dots +R_1·R_2···R_{n-1}}{R_1·R_2···R_n}}$$

where R_T is the total charge transfer resistance, and R_i is the resistance of i-th ionic current path. Now, considering that R₁ = R₂ = ⋯ = R_n without loss of generality, we obtain a rational function for total charge transfer resistance that decreases with current:

$${\displaystyle \frac{1}{R_T}}={\displaystyle \frac{n·R^{n-1}}{R^n}}\to R_T={\displaystyle \frac{R}{n}}\to R_T={\displaystyle \frac{R}{n\left(I\right)}}$$

In light of this, $\mathcal{V}$, b, and d are the fitting parameters of a rational function to describe the behavior of $R_{tc}$ with current. Then, according to our interpretation, d would be related to the existence of some tiny residual current that is independent of the power demand, and the parameter b is used to fit the dependence of $R_{tc}$ with current. Because the resistance of all ionic paths has not to be equal, b may be a measure of the double layer homogeneity, representing b = 1 as a homogeneous double layer. It is convenient to notice that the $R_{tc}$ behavior modeling is not a closed issue, and other mathematical prototypes can be checked. We use Equation (2) for its simplicity and because it works.