Application of Buckingham p theorem for scaling-up oriented fast modelling of Proton Exchange Membrane Fuel Cell impedance

(cid:1) A methodology to reproduce PEMFC impedance is proposed. (cid:1) Non-dimensional parameters are de ﬁ ned by exploiting the Buckingham's p theorem. (cid:1) Good accuracy in PEMFC impedance prediction is proved. (cid:1) The possibility to use this methodology with scaling-up purposes is demonstrated. This work focuses on the development of a fast PEMFC impedance model, built starting from both physical and geometrical variables. Buckingham's p theorem is proposed to de ﬁ ne non-dimensional parameters that allow suitably describing the relationships linking the physical variables involved in the process under-study to the fundamental dimensions. This approach is a useful solution for those problems, whose ﬁ rst principles-based models are not known, dif ﬁ cult to build or computationally unfeasible. The key contributions of the proposed similarity theory-based modelling approach are presented and discussed. The major advantage resides in its straightforward online applicability, thanks to very low computational burden, while preserving good level of accuracy. This makes the model suitable for several purposes, such as design, control, diagnostics, state of health monitoring and prognostics. Experimental data, collected in different operating conditions, have been analysed to demonstrate the capability of the model to reproduce PEMFC impedance at different loads and temperatures. This results in a reduction of the experimental effort for the FCS lab characterization. Moreover, it is highlighted the possibility to use the model with scaling-up purposes to reproduce the full stack impedance from single- cell one, thus supporting FC design and development from lab-to commercial system-scale.


Introduction
Among the environmentally friendly energy conversion technologies, one of the most suitable solutions to cope with pollutant emissions and global warming issues are fuel cells, which have attracted the interest of many researchers in the last decades. In particular, the Proton Exchange Membrane Fuel Cell (PEMFC) has proved to be the most suitable for applications between 0.1 and 1000 kW [1]. The main advantages of PEMFCs are the high performance, modularity and potentially pollution-free operation. However, this technology has to meet specific technical requirements to become commercially competitive. US Department of Energy (DOE) and European Institutions have set clear technical targets to be achieved by PEMFC systems with costs and lifetime being the main issues; for stationary and automotive applications 40.000 h and 6000 h of operations are required, respectively ( [2,3]).
In order to enhance FCS reliability and lifetime, control and diagnostics must be improved thanks to advanced monitoring algorithms, which must be fast enough to operate on-field to obtain effective advantages. Advanced techniques are required to analyze performance losses caused by degradation mechanisms and the impact of the current operating conditions.
The most effective methods used to characterize the performance of a fuel cell are electrochemical techniques, such as Electrochemical Impedance Spectroscopy (EIS), polarization curves, cyclic voltammetry, AC impedance and current interruption techniques [4]. Among these techniques, EIS is the most suitable to recognize the individual contributions related to voltage drops caused by membrane, charge transfer and mass transfer resistances, both for single cells and overall stack [5]. The EIS technique is used to measure the frequency dependence of the impedance of a fuel cell by applying a small sinusoidal AC voltage (or current) as a perturbation signal to the fuel cell and measuring the current (or voltage) response. This is done at different frequencies for each load to obtain the overall system impedance, often plotted as Nyquist diagram [6]. The applications of EIS in PEMFC studies allows providing electrochemical information about the FCS, which can then help in fuel cell structure optimization and selection of the most appropriate operating conditions. The EIS appears to be the most suitable method for fuel cells on-field monitoring and control, thanks to its peculiar characteristics [7]. Indeed, the main advantage of using EIS resides in the possibility to recognize the contributions of each physical phenomenon (i.e. ohmic losses, charge-transfer and diffusion limitations) to the full impedance thanks to the Equivalent Circuit Modelling (ECM), which consists in modelling the cells using equivalent circuits. Therefore, each contribution can be identified since each circuital parameter represents a physical phenomenon occurring inside the fuel cell [8]. Moreover, the analysis of the ECM parameters may be useful to distinguish between normal and faulty operations [9]. The use of Fractional Order Modelling (FOM), instead, has been proposed as a very suitable approach to model transients and time depending phenomena [10].
Many authors have demonstrated the capability to detect faults using the EIS by recording impedance spectra during fuel cell operation and analysing these data using the ECM approach with diagnostic purposes ( [11,12]). The paper of Narjis et al. [13] provides the main references for the development of Fault Detection and Isolation (FDI) based on EIS [7]. Also the use of more physical-based modelling representation of fuel cell impedance has been thoroughly accounted for in the literature. For example, Kulikovsky proposed the analytical model of a the impedance of a PEMFC cathode catalyst layer under poor oxygen transport ( [14,15]) and at Open Circuit Voltage in addition to the Gas Diffusion Layer impedance [16]. The work of Cruz-Manzo et al. [17] describes the PEMFC impedance spectra subject to Pt oxidation and H 2 peroxide formation at cathode side, also with the combined use of physical and circuit modelling. On the same line, Cruz-Manzo et al. [17] and Niya et al. [18] developed an impedance model combining physical and ECM approaches, particularly aiming at describing a PEMFC impedance at different operating conditions, varying current densities, temperature and humidity levels. Physics-based PEMFC impedance models were also developed by Setzler and Fuller [19], with the implementation of an oxide growth oxygen reduction reaction kinetic model, and Chevalier et al. [20,21], aiming at PEMFC state-of-health and degradation analysis. The analysis of the cells Output Voltage Signal with Discrete Wavelet Transform (DWT) and Continuous Wavelet Transform (CWT) has been also proposed for fuel cells FDI in the recent literature ( [22,23]).
The EIS methodology, however, has its drawbacks. The traditional EIS equipment is costly and cumbersome and requires several minutes to record a single impedance spectrum. Moreover, the diagnosis can be performed once some metrics are derived through EIS signal treatment. This is not a simple task because, in many cases, different operating conditions could lead to the same impedance spectrum; thus, a deep analysis based on analyst experience is necessary. On the other hand, for on-field (i.e. unattended) operations suitable algorithms, such as those based on the ECM [24] or on FOM approaches [10], can be built to process the EIS to derive useful metrics for monitoring and diagnostic applications.
The scientific contribution of the present work is the development of a fast model capable of simulating PEMFC impedance spectra. The model is based on non-dimensional parameters defined by exploiting the Buckingham's p Theorem, which is a key method in the similarity theory field. Particularly, it allows suitably describing the relationships linking the physical variables involved in the process under-study to the fundamental dimensions [25]. The definition of non-dimensional parameters allows reproducing the impedance of different types of system operating at different conditions, thanks to the high generalizability of the model. Therefore, the development of such a model does not require large experimental data; moreover, its simple algorithms need few coding operations, low experimental efforts and are computationally efficient (i.e. fast), allowing easily implementation on low cost hardware as well. This method may be then used for several purposes either for design purposes, by exploiting its scaling-up features, or on-system implementation for on-field use such as on-line applications (e.g. monitoring, diagnostics, control, state of health management and prognostics). The authors have already discussed the significant capabilities associated to the proper exploitation of similarity theory for optimal design and management of innovative energy systems, including fuel cell vehicles ([26e28]). The scalingup features of the presented method allow using either single-cell or short stack arrangements to characterize the electrochemistry, so as to simulate the behaviour of the full stack. During prototyping phases, this may help in reducing hardware costs (for both test bench and cell materials), as well as consumables. Moreover, it is also beneficial in those problems where the occurrence of degradation and faults should be tested under different operating conditions, i.e., when non-optimal control variables are imposed. It is a matter of fact that to characterize FCS several experimental tests have to be performed under different loads. Moreover, to be able to identify occurring faults, tests in faulty operating conditions have to be performed too. These tests are very degradative and implies FCS fast end-of-life. Thanks to the proposed similarity theory-based modelling approach, the latter issues can be avoided by performing a unique EIS measurement at fixed operating conditions, thus obtaining the behaviour of the FCS under different operating conditions by re-characterizing the non-dimensional parameters.
The model developed following the Buckingham theory is mainly achieved through a data-driven approach, which links measured impedance data and physical models to shape the nondimensional parameters required for the problem description and generalization.
In the following, the description of the overall model is firstly given, starting from the definition of the non-dimensional parameters towards the description of each involved variable. The design problem is formulated considering both physical and electric variables, which are representative of fuel cell impedance (as detailed in Section 2.1). Therefore, key features of both ECM-based and physics-based methods are accounted for. Nevertheless, it is worth observing that the characterization of these non-dimensional parameters requires the use of impedance data provided by experimental tests. The use of such data allows shaping the fuel cell impedance through the non-dimensional parameters by means of proper physical variables (e.g. temperature, current, water content, fuel utilization, etc.), whose values depend on the specific conditions that have to be simulated. Therefore, the novelty of this approach resides in its structure and development process, which could be considered as black-box if seen from a mathematical point of view (as could be any polynomial regression). However, as said, it gathers properties of both ECM-based and physical-based representations: the impedance layout is given by the measured data required for non-dimensional parameters evaluation (without requiring any ECM model), and the scaling-up process is fulfilled through physical reasoning (e.g. inference on fuel and/or water distribution or other significant parameters). According to these comments, the main advantage of using such an approach, with respect to ECM or physics-based methods, consists in the easy and quick impedance scalability, achieved through proper scalability factors (as for instance those used in other disciplines like fluidand thermal-dynamics through Reynolds and Prandtl numbers), without requiring high computational processes or complex models. This gives further degree of freedom in the impedance representation, without requiring the definition of, e.g., a specific impedance or model shapes. Afterwards, the paper deals with the application of the methodology both to reproduce PEMFC impedance at different operating conditions and simulate PEMFC impedance from single-cell to full stack (i.e. scaling-up). This study is supported by the analysis of experimental data available in the literature ([29e31]) which were particularly deployed to assess model accuracy as well as the scaling-up potentialities of the proposed methodology. To authors knowledge, no previous contributions were given on the application of Buckingham theorem to PEMFC impedance shape reproduction.

PEMFC impedance model
The idea of developing a fast PEMFC impedance model arises since the most suitable method used in FCs diagnostics and control is based on EIS measurements. However, the current methodology, that involves the analysis of the measurements through circuital modelling, might be time consuming and would require further work for its on-line application within FDI algorithms [6]. On the other hand, the model proposed in this study has the advantage of being computationally efficient and easily generalizable, thanks to the exploitation of non-dimensional parameters obtained via a dimensional analysis of the involved physical quantities.

Dimensional analysis
The developed model uses non-dimensional parameters to reproduce the PEMFCs impedance. To define these parameters, the Buckingham's p theorem has been exploited ( [25,32,33]). This approach has been previously used to address both engineering [34] and electrochemical [35] problems. The theorem states that if there is a physically meaningful equation involving a number n of physical variables, then the original equation can be rewritten in terms of a set of non-dimensional parameters p 0 , p 2 , …, p p-1 built from the original variables. Let be Q 0 a physical variable of interest that is a "dependent variable" in a known process. This means that once all the independent quantities are specified the value of Q 0 follows uniquely. The procedure to identify the non-dimensional parameters that can simplify and generalize the problem is the following [36]: Step 1: Identify a complete set of independent quantities Q 1 , Q 2 , …, Q n that allow determining the value of Q 0 : Step where the exponents l i , m i , t i and j i are dimensionless numbers that follow from each quantity's definition and i ¼ 0, n.
Step 3: Pick from the entire set of independent variables a "complete" and "dimensionally independent" subset and express the dimensions of the remaining variables, including the dependent variable Q 0 , as a product of powers of this subset. A subset is "dimensionally independent" if none of the variables included has dimensions that can be expressed in terms of the dimensions of the remaining variables included. It is "complete" if the dimensions of all the remaining quantities outside the subset can be expressed in terms of the dimensions of the subset. The determination of this subset is driven by heuristic or trial and error approaches. It involves k variables, with k v, where v is the number of involved fundamental dimensions.
Step 4: Define the non-dimensional parameters. The number of non-dimensional parameters is evaluable as: Afterword, the non-dimensional parameters are obtained by dividing the variables not included in the previously defined subset with the product of the variables included in the subset: where N (i)1 , N (i)2 , …, N (i)k are the powers that zero the dimensions and i ¼ 0, (p e 1). The result is a simplification of the overall problem, mainly due the reduction of the number of involved quantities. Indeed, the problem can be rewritten as: where p 0 is the non-dimensional parameter defined from the dependent variable Q 0 . The PEMFC impedance can be defined as a function of specific variables, either physical [37] or electrical [38], used to describe the electrochemical phenomena that take place inside the cells. Therefore, the PEMFCs impedance is here defined as a function of both electrical and electrochemical variables: where R eq represents the equivalent electric resistance, C the electric capacitance, u the frequency, I the electric load and ECSA the electrochemical surface area. The equivalent electric resistance describes the linear losses (e.g. membrane resistance and chargetransfer resistance) while the electric capacitance shapes the non-linear ones (e.g. energy accumulation over the electrodes). Therefore, after Eq.

> > > > > > <
> > > > > > : Following the procedure described in Step 3, the complete and dimensionally independent subset is defined: it includes the variables R eq , u, I and ECSA. Thus, the number of non-dimensional definable parameters to describe the overall problem, according to Eq. (3), is two (n ¼ 6 and k ¼ v ¼ 4). The first is that one obtained from the dependent variable Z, the second one is obtained from the independent variable C. Following the aforementioned procedure: As long as the objective is to obtain non-dimensional parameters, the following system has to be solved to obtain the powers N i : So, the unique value different from zero is N 01 . Thus, the first non-dimensional parameter is: Following the same procedure for the variable C: In this case, the system to be solved is: leading to the second parameter: The first parameter is the normalization of the total impedance over the equivalent resistance, while the second one is a time dependent group. The two non-dimensional parameters obtained are referred, from now on, as G and k, respectively. The outcomes of the above-described dimensional analysis underline that two nondimensional parameters are necessary overall to define the impedance. Since the total impedance involves real and imaginary parts, it is possible to distinguish between real and imaginary parts of G, defined as real and imaginary parts of the total impedance normalized over the equivalent resistance, as shown in Eq. (14): Thanks to this simplification, to reproduce impedance spectra in the Nyquist diagram at different operating conditions or for different system sizes, only the parameters G r and G i are needed. These parameters can be identified from EIS experimental data. Indeed, the variables Re(Z) and Im(Z) are known data coming from EIS measurements (i.e. real and imaginary part of the cell impedance and vector of the scanned frequencies, respectively). The equivalent resistance, indeed, is defined by exploiting suitable models, as described in the next section.

Variables modelling
Thanks to the dimensional analysis and the application of the Buckingham's Theorem, the non-dimensional parameters capable to simplify and generalize the problem have been defined in Eq. (14). Fig. 1 shows the effective steps, to be undertaken to fruitfully exploit the proposed approach for PEMFC impedance modelling, as well as for scaling purposes.
In the identification phase, Re(Z) and Im(Z) (i.e. real and imaginary parts of the impedance) are derived from an existing EIS data set and, thanks to the physical and empirical models detailed below, within this section, R eq is computed, so that the nondimensional parameters G r and G i can be successfully identified (see Eq. (14)).
Then, to obtain impedance behaviour in different operating conditions or for different system size, in the scaling phase R eq has to be recomputed in accordance with the conditions to be predicted (e.g. if the objective is to predict impedance behaviour in different load conditions, the value of the load I must be changed, whereas if the objective is to scale the impedance to higher power systems, geometrical parameters have to be adjusted, etc.). The scaling procedure will result in a new value of the total resistance R Ã eq , as well as of the real and imaginary parts to be predicted, i.e. Re * (Z) and Im * (Z), as shown in Eq. (15): The equivalent resistance R eq is the sum of ohmic (R m ) and charge-transfer (R ct ) losses. This results from having assumed the mass transfer contribution negligible because, at this moment, the overall model mainly aims at reproducing PEMFC impedance spectra in normal operating conditions, whereas, as it is wellknown, mass transfer losses are usually associated to abnormal operating conditions (e.g. if the stack were operated at loads higher than what is recommended by the manufacturer, an accentuated mass transfer arc would occur in the impedance spectrum). However, for future prospects (i.e. PEMFC diagnosis and prognosis), it has been considered by the authors the addition of the mass transfer losses related variables (such as H 2 inlet pressure), so as to make the overall model robust also in case of abnormal operating conditions.
Thus, the equivalent resistance is: The ohmic resistance can be computed using the model proposed by Springer [39]: where l m is the membrane thickness, A is the electroactive area, and s m is the membrane conductivity, expressed as: From Eq. (18), it is possible to note that the conductivity is expressed as a function of the mean temperature of the stack (T S ) and the parameter s 30 , which is the conductivity at 30 C defined as function of the membrane humidification [39]: Alternatively, the membrane resistance can be expressed by exploiting the model of Mann [40]: In this case, the membrane resistance is expressed as a function of the current density (I/A), the cell temperature expressed in Kelvin degrees (T S ), the membrane humidification level l m and the parameter K m , related to the membrane thickness. The membrane humidity level l m is function of the relative humidity (RH), which represents a clear indicator of the amount of water that wets the membrane. It is defined as the ratio of the partial pressure of the water vapour to the equilibrium vapour pressure: Considering Nafion ® 117, as long as this type of membrane has been largely adopted in PEMFC applications (and thus in the literature a large amount of studies is available that allows obtaining the relationship between RH and l m , useful in modelling ohmic losses ([39e41])), membrane humidification reaches the value l m ¼ 14 at a relative humidity of 1. At this condition, the ohmic resistance assumes the smallest possible value as demonstrated so far by Springer [39], and confirmed by Kawamura [41], who has investigated further operating conditions. For values of l m < 14 (membrane drying) the ohmic resistance increases; for values of l m > 14 (membrane flooding), instead, the ohmic resistance remains the same, but the high presence of water at both sides could cause reactants starvation, as well as an increase in diffusive transport.
The data collected by Kawamura [41] have been used to obtain a polynomial function that correlates the membrane humidification and RH as follows: l m ¼ 35:9$RH 3 À 39:6$RH 2 þ 17:7$RH þ 0:0441 (22) It is worth clarifying that, for different technologies, it is possible reproducing the ohmic losses by changing membrane conductivity in the variables models, in accordance with the material of the membranes.
The relative humidity value has been fixed at the stack middlecell and it has been considered equal to 1 for the closed-cathode stacks and 0.6 for the open-cathode ones, because it is well known that this type of stacks works in dryer conditions as it is characterized by a very high airflow that dries the membrane on the cathode side moving the water outside faster than it is created. Moreover, it has been hypothesized that RH has linear behaviour increasing/decreasing by 2% per cell going toward air and fuel inlet, respectively. This is an empirical hypothesis consequence of the well-known water distribution along the stack, from the H 2 inlet to the Air/O 2 one. Indeed, the cells closer to the Air/O 2 inlet, show a higher water content compared to the cells closer to the H 2 inlet. Also for the Fuel Utilization (FU), a linear distribution has been adopted: FU increases linearly from the 1st cell till the last one (from 0.1 to 1) (Fig. 2). The linear behaviours have been adopted for simplicity and to make the algorithm (that has real-time applications purposes) faster; the overall precision, shown in the following results section, is the proof of concept that this hypothesis is not so far from the real behaviour of the water distribution inside the stack.
The charge-transfer resistance (R ct ) is expressed through a regression obtained by exploiting experimental data collected in different operating conditions. In particular, R ct is expressed as a function of the current, temperature, electrodes area and number of cells (Eq. (23)). The regression is obtained from data collected in the framework of the European Project D-CODE [29]. In particular, the ECM algorithm proposed by Petrone in his PhD Thesis [42] has been exploited to obtain the value of the charge transfer resistance at different operating conditions; such values were then regressed as a function of stack mean temperature expressed in Celsius degrees (T S ) and stack current load I, thus resulting in the following correlation: It is worth remarking that the procedure implies the identification of the non-dimensional parameters behaviour on a set of EIS experimental data. Once the polynomials that define the real and imaginary parts of G as function of k are identified, these parameters have to be re-characterized through the aforementioned physical and mathematical models to obtain impedance spectra for different system size (i.e. scaling-up) or at different operating conditions (e.g. higher current loads).
As mentioned in the previous sections, the proposed approach requires the use of measured impedance data to characterize the non-dimensional parameters of the Buckingham theorem. The shape of the data used in this task reflects part of the scaled impedance shape: if the measured impedance shows a depressed half circle, the scaled up ones may show it as well. However, the scaling up process also plays an important role through the introduction of proper values of the accounted physical parameters. The proposed model involves both electric and physical variables and represents an extension of the ECM approach. Differently from a simple ECM that uses only electric parameters, the proposed approach allows linking the phenomena that take place inside the cells to the correspondent physical variables (e.g. RH, FU, electroactive area, membrane conductivity, etc.). It is precisely thanks to the physical formulation of the involved variables (i.e. modelling R m with Springer or Mann laws, modelling R ct as a function of current and temperature, etc.) that it results possible to reproduce the impedance behaviour of the cells in different operating conditions and for different systems size. Indeed, the suitable definition of fuel or water content distribution (as introduced in the following) allows catching impedance deformation due to, e.g. faults or degradation processes. The present algorithm cannot thoroughly represent all the possible shapes a fuel cell impedance spectrum may show.
Indeed, thanks to this approach, that exploits the advantages of the similarity theory combined with physical models of the variables, it is possible to reproduce accurately the PEMFC impedance. In the next section, the capability of the model to reproduce PEMFC impedance both in different operating conditions and for different stack sizes is shown.

Results
The proposed model allows obtaining the PEMFC impedance in different operating conditions and could be used with different purposes. In this section, the capability to reproduce the impedance starting from a unique set of experimental data is shown. The last section of the chapter is dedicated to the scaling-up. The scaling-up has, by definition, the objective to migrate a process from the labscale to the pilot plant-scale or commercial scale [44]. During the design phase, it is common use performing degradative tests on entire FCS to characterize these systems and their behaviour operating in faulty conditions. These tests are very expensive as they imply fast end-of-life of the systems. It is possible to avoid this problem by testing single-cells to get their behaviour under faulty operating conditions and, then, scaling-up the results to the whole fuel cell systems.

PEMFC impedance at different current loads
To characterize FCS, many tests in different operating conditions should be performed. To reduce the number of tests and, thus, time and cost efforts, the proposed methodology can be adopted. In this way, only one test at a fixed current load is needed to fully characterize the FCS, provided that the extension of the black-box model, here proposed for charge transfer resistance estimation (see Eq. (23)), is proven valid. To validate the model capability to predict PEMFC impedance spectra in different operating conditions, the experimental data collected in the framework of the D-CODE European project [29] and those one available from the paper of Asghari [30] have been used. In Table 1 are collected the main characteristics of the stacks tested in the framework of these works. It is worth remembering that the PEMFC impedance spectrum, when represented on the Nyquist diagram, is characterized by one arc related to the cathodic losses. Indeed, the losses related to anode and mass transport limitations are negligible in normal operating conditions. Thus, as the current load increases, the PEMFC impedance arc shrinks [1].
In Figs. 3 and 4 the results in reproducing the impedance spectra at different currents for the two datasets are shown. The nondimensional parameters (Eq. (14)) have been identified on the data acquired at 5 A and 20 A for D-CODE and Asghari, respectively (see Table 1). These have been than scaled, following the procedure described in the previous section (Fig. 1), to obtain the spectra at 10, 25 and 40 Amps for D-CODE (Fig. 3), and 40 and 60 Amps for Asghari (Fig. 4). The experimental data are depicted with empty markers, whereas model results are depicted with filled markers; the lines colours are similar for equal loads. The impedance has been normalized multiplying by the area and dividing by the number of cells. At first glance, a good accordance emerges between the experimental data and the model predictions.
Focusing on Fig. 3, it is possible to note that the precision of the model predictions decreases as the load increases. This could be due to the fact that the temperature of the stack changes with the current, while in the model it is fixed at the value assumed at 5 A, because in the real application of the proposed methodology the temperature at higher loads may be unknown. Indeed, for the data of Asghari (Fig. 4), which have been acquired at fixed temperature (i.e. 55 C), the precision is higher for all the loads. Anyway, the precision is high for all the analysed cases and it is comparable to that one obtainable by using equivalent circuit modelling [7]. This confirms the possibility to perform less laboratory tests to fully characterize stacks behaviour under different loads conditions, mainly thanks to the good accuracy guaranteed by the here proposed similarity theory-based model to reproduce PEMFC impedance shape. It is finally worth remarking that the charge transfer resistance model (i.e. Eq. (23)) was identified on the D-CODE data set [29]. Afterwards, such a sub-model was directly extended, with minor adaptation, to Ashgari data set [30], thus fully demonstrating the genericity and the potential technology independent deployment of the proposed procedure.

Scaling-up of PEMFC impedance: from single-cell to full stack
The scaling-up of fuel cells is usually performed after the singlecell has successfully met its targets of durability and reliability, including all the components in the cell, such as membrane, GDL, CL, BP and electrodes. However, the performance of the stack or of the entire FCS are difficult to derive from the single-cell data. Wang ([43,44]) in his studies describes this issue and the necessity to compute and optimize the flows distribution thanks to flow field design that needs CFD analysis. Paliniswamy [45] have conducted similar studies with the objective to scale-up PEMFC with different flow fields from 25 to 70 cm 2 . In the work of Bonnet [46], instead, a comparison between a 25 cm 2 single-cell, a 340 cm 2 /5-cell pilot stack and 800 cm 2 /90-cell stack is shown to define the best operating conditions, yielding the highest performance and the lowest degradation on the complete 80 kW FCS. Despite to sophisticated and time-consuming methodologies based on CFD analysis, the possibility to scale-up electrochemical systems by exploiting Similarity Theory thanks to the definition of non-dimensional parameters is discussed in the works of Rao [47] and Sulaymon & Abbar [48]. The data analysed to validate the scaling-up capability of the proposed method are those collected by Wasterlain [31]. These data include the impedance spectra of a 20 cell PEMFC stack (with electrodes area of 100 cm 2 ) recorded both for each cell and for the overall stack. According to the procedure described in the previous section ( Fig. 1), from the treatment of the impedance data of a single-cell the non-dimensional parameters G and k have been derived. Once the polynomials defining the real and imaginary parts of G as function of k had been computed, these parameters have been re-characterized to obtain the overall impedance of the stack.
As it can be noted, the last-cell is characterized by higher diffusion losses. This can be due to the flow distributions in the stack. Indeed, the last cell is the closest to the air inlet and the fuel outlet: the accumulation of water in this zone is an obstacle to the hydrogen flow [31]. Fig. 5 provides an explanation of the two different analyses done by exploiting the data collected by Wasterlain et al. [31]. Particularly, the first analysis was carried-out by identifying the non-dimensional parameters on the stack middlecell and then scaling R eq to obtain the stack total impedance; the second analysis, instead, was performed by identifying the nondimensional parameters on the stack last-cell and then scaling R eq to obtain the stack total impedance.

From stack middle-cell to whole stack impedance
In Fig. 6 the comparison between the results obtained by exploiting the models of Springer (Eq. (16)) and Mann (Eq. (19)) for the membrane resistance are shown. To identify the nondimensional parameters, the data related to the stack middle-cell have been considered. These data have been used to reproduce the stack impedance. From the comparison of the stack experimental data and the models predictions, a good accordance emerges. The comparison of the percentage errors of the models predictions, by using Mann and Springer models for the membrane resistance, with respect to the experimental data are shown in Table 2. Here, the percentage errors to predict R m (that is the impedance arc 1st intercept on the real axis), R ct (that is the 2nd intercept on the real axis) and the maximum value of the impedance imaginary part (absolute value) are shown. The resulting accuracy is similar and the use of one rather than the other is related to the scope of the analysis. If the objective is to migrate experimental data collected with a PEMFC made with Nafion 112 to a stack made with different materials, for instance, the model of Springer is useful because it involves the dependence by the membrane conductivity. Fig. 3. Results in PEMFC impedance prediction at different operating currents (D-CODE data [29]). The experimental data are depicted with empty markers, whereas model results are depicted with filled markers. Fig. 4. Results in PEMFC impedance prediction at different operating currents (Asghari data [30]). The experimental data are depicted with empty markers, whereas model results are depicted with filled markers.

From stack last-cell to whole stack impedance
In Fig. 7, instead, the results obtained scaling the impedance of the stack last-cell are reported. The non-dimensional parameters have been identified by exploiting the EIS data related to the worst cell in terms of water content (in this case the closest to the air inlet, that results flooded). This choice has been done to validate the model capabilities and its robustness: indeed, the simple multiplication of the impedance of this cell by the number of cells (Single-cell*Ncell) does not allow obtaining good prediction of the overall stack impedance, while by using the proposed methodology, even if the non-dimensional parameters have been identified using the more flooded cell, the total stack impedance prediction (Scaling-up) is in good accordance with the value measured experimentally (Stack exp).
Here is also reported the spectrum obtainable with the simple multiplication of the single-cell impedance times the number of cells to highlight that it is not suitable this simple multiplication to obtain the full-stack impedance; in fact, this multiplication implies an overestimation of the stack total impedance, while the model allows obtaining a good prediction. Indeed, the results obtained for the full-stack impedance prediction have a precision comparable to equivalent circuit methods [7] with a mean percentage error, calculated by exploiting Eq. (24)e(25), of 10 and 20% to predict real and imaginary parts, respectively.
PÀ ReðZÞ À Re * ðZÞ Á ReðZÞ n p $100 PÀ ImðZÞ À Im * ðZÞ Á ImðZÞ n p $100 (25) In Table 3 are collected the non-dimensional parameters G r and G i defined through the proposed approach for the different analyses shown in this section.

Conclusions
This work presents a methodology to reproduce PEMFC impedance shape thanks to non-dimensional parameters defined by exploiting the Buckingham's p Theorem, a key method in the similarity theory field. The procedure implies the identification of the non-dimensional parameters on a unique set of experimental data. After that, by re-characterizing these parameters, it is possible to reproduce PEMFC impedance in different operating conditions (e.g. current loads, operating temperatures, membrane humidification level, etc.) and for different system size (i.e. scaling-up). The capabilities of the method have been validated on data present in literature by comparing the model results and the experimental data. The precision of the model is comparable to that one obtainable using equivalent circuit modelling approach. Thanks to the proposed methodology, it could be possible to reproduce the impedance of a FCS completely different from that one used to identify the non-dimensional parameters (in terms of number of cells, electrodes surface, membrane material). Moreover, it could be possible to reduce the number of tests to be performed to fully characterize the FCS under different operating conditions, as the   proposed approach allows accurately reproducing real impedance curves acquired in operating conditions falling out of the domain considered in the model development phase. Regarding the latter aspect, moving from black-box to a more physical modelling of the charge transfer resistance dependence on main operating variables (i.e. temperature and current) is expected to further strengthen the genericity features of the proposed scaling-up method. The proposed approach can be used with several purposes such as design, diagnostics, prognostics, state of health management and on-board control, as it allows obtaining PEMFC impedance in different operating conditions with good precision and low computational effort. 256673 (project D-CODE e DC/DC converter-based diagnostics for PEMFC systems) and from the Fuel Cells and Hydrogen 2 Joint Undertaking under grant agreement No 671486 (project HEALTH-CODE -Real operation pem fuel cells HEALTH-state monitoring and diagnosis based on dc-dc COnverter embeddeD Eis). This Joint Undertaking receives support from the European Union's Horizon 2020 research and innovation programme and Italy, Denmark, Germany, France.