A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

Available online at www.sciencedirect.com

ScienceDirect journal homepage: www.elsevier.com/locate/he

Short Communication

A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells In-Su Han a, Chang-Bock Chung b,* a b

R&D Center, GS Caltex Corp., 359 Expo-ro, Yusung-gu, Daejeon 34122, South Korea School of Applied Chemical Engineering, Chonnam National University, Gwangju 61186, South Korea

article info

abstract

Article history:

A hybrid model was proposed by combining a support vector machine (SVM) model with an

Received 11 October 2016

empirical equation for more accurate prediction of the polarization curves of a PEM

Received in revised form

(polymer electrolyte membrane) fuel cell under various operating conditions. Operational

18 December 2016

data were obtained from designed experiments for a PEM fuel cell for training, testing, and

Accepted 23 January 2017

validating the hybrid model, and a model training procedure was presented for deter-

Available online xxx

mining the model coefficients and hyper-parameters of the hybrid model. The predictive performance of the hybrid model was compared with that of a SVM model. The SVM model

Keywords:

showed somewhat poor performance, especially yielding large prediction errors in the high

Polymer electrolyte membrane

voltage ranges of the polarization curves as reported in the literature. In contrast, the

(PEM) fuel cell

hybrid model exhibited almost perfect matches between the predicted and measured po-

Polarization curve

larization curves, resulting in significantly lower root-mean-square errors of 1.7e4.4 mV

Data-driven model

which correspond to only 14e21% of those obtained from the SVM model.

Support vector machine (SVM)

© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved.

Hybrid modeling

Introduction Fuel cells have been gaining increasing attention because of their many advantages over conventional energy conversion systems such as internal combustion engines and gas turbines. Numerous studies have been performed and published, including the modeling and simulation studies which played important roles in the prediction, analysis, diagnosis, and optimization of fuel cell operation. The developed models covered not only mechanistic models describing the

physicochemical phenomena taking place in the fuel cells [1e3] but also data-driven (or black-box) models based on functional relationships between the input and output variables of fuel cells [4e9]. The performance of a fuel cell is usually characterized by measuring a polarization curve (also called ieV curve) that can be obtained by plotting current density (or current) versus cell potential. A polarization curve definitely depends on various operating variables of a fuel cell such as current, temperature, pressure, flow rate, and relative humidity, and its shape can be fitted using various types of mathematical models,

* Corresponding author. Fax: þ82 625301909. E-mail addresses: [email protected] (I.-S. Han), [email protected] (C.-B. Chung). http://dx.doi.org/10.1016/j.ijhydene.2017.01.131 0360-3199/© 2017 Hydrogen Energy Publications LLC. Published by Elsevier Ltd. All rights reserved. Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131

2

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

including mechanistic [10e12], empirical [13e18], hybrid [19,20], and data-driven models based on machine learning techniques [21e25]. Among these models, data-driven models based on artificial neural networks (ANNs) or support vector machines (SVMs) have been mostly employed because of their great flexibility in determining the model structures and good predictive performance. Several papers reported that ANN models fitted well to the polarization curves with satisfactory accuracy [21,22]. However, some previous studies demonstrated that SVM models produced considerable deviations between predicted and measured polarization curves, especially in the high voltage ranges of the polarization curves [24,25]. In this study, we propose a hybrid model, which combines a SVM model with an empirical equation, for more accurate prediction of the polarization curves of a PEM (polymer electrolyte membrane) fuel cell under various operating conditions. The proposed hybrid model is expected to compensate for the prediction errors of the SVM model.

Model structure The hybrid model proposed in this study consists of two submodels as shown in Fig. 1: the empirical equation (model) and the SVM model. The empirical equation accounts for the b emp ) owing to the major variations in the cell voltage ( V changes in the current density (i), thus capturing the overall shape of a polarization curve. In contrast, the SVM model explains the minor variations in the cell voltage owing to variations in the other operating variables such as temperature, pressure, and relative humidity as well as the current density, thus compensating for the mismatch (DV) between the predicted and measured polarization curves. More specifically, the SVM model predicts the voltage difference b svm ) between the measurement (V) and the prediction (D V emp b (V ) from the empirical equation as a function of various operating variables (i and x1, x2, …, xv). Then, the final preb of the hybrid model is calculated by dicted cell voltage ( V) b svm from the SVM adding the predicted voltage difference D V emp b from the empirical model to the voltage prediction V equation:

b ¼V b emp ðiÞ þ D V b svm ði; x1 ; x2 ; /; xv Þ V

(1)

b emp from the In the above equation, the voltage prediction V empirical equation can be described by the following equation: b emp ¼ E  hact  hohmic  hconc V

(2)

where E is the theoretical reversible open circuit voltage, hact is the activation over-potential expressed by the Tafel equation hact ¼ a þ b ln (i) (written here in the natural logarithmic form), hohmic is the ohmic over-potential expressed by hohmic ¼ Ri, and hconc is the mass transport over-potential expressed by hconc ¼ p exp(qi). Rearranging Eq. (2) after replacing the over-potential terms with their detailed expressions, we can get the following empirical equation: b emp ¼ fðiÞ ¼ E0  b ln ðiÞ  R i  p exp ðq iÞ V

(3)

where E0 represents Ea. This equation was suggested by Kim et al. [16] and has been frequently employed to fit the polarization curves of fuel cells at constant operating conditions [16e18,21]. Because the current density is the only input variable of the equation, the five coefficients (E0, b, R, p, and q) must be determined anew using a nonlinear parameter estimation method whenever there is a change in operating conditions under which the polarization curve is measured. The SVM model was constructed using a standard support vector regression scheme (also called ε-SVM) as proposed by Vapnik et al. [26]. In the context of our problem, given a training data set of n samples consisting of an input matrix of the operating variables X ¼ [x1, x2, …, xn] and an output vector of the voltage differences DV ¼ ½DV1 ; DV2 ; /; DVn , the unb svm and x was approximated known functionality between D V by the linear combination of a finite number of coefficients (a and a*) and a kernel function K(x, xi) as follows: b DV

svm

¼ gðxÞ ¼

n X 

 ai  ai Kðx; xi Þ þ r

(4)

i¼1

where a and a* are the Lagrange multipliers, and r is the bias term. The kernel function K(x, xi) maps the input space into a high dimensional feature space. In this study, a radial basis

Fig. 1 e Structure of the hybrid model consisting of a SVM model and an empirical equation. Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

function (RBF) kernel expressed by Kðx; zÞ ¼ expðskx  zk2 Þ was employed which had been reported to produce good predictive performance for regression problems [7,27]. The vectors of the Lagrange multipliers a* ¼ ½a1 ; a2 ; /; an  and a ¼ ½a1 ; a2 ; /; an  were found by solving the following quadratic programming (QP) problem with constraints: n n n X     X   1X  DVi ai  ai þ ai þ ai  minL a* ; a ¼ ε 2 i¼1 i¼1 i¼1



n X 

ai  ai

     aj  aj K xi ; xj

n  P i¼1

respectively). Besides these data sets, a validation data set was prepared from two more stack operations under the following conditions distinct from those for the training and testing data sets.: OP1) PH2 ¼ 230 kPa, PO2 ¼ 230 kPa, TS ¼ 70  C, and RH ¼ 80% and OP2) PH2 ¼ 150 kPa, PO2 ¼ 150 kPa, TS ¼ 50  C, and RH ¼ 40%. Each data set was normalized to the range 0e1 using a minemax normalization method before use for its respective purpose.

(5)

Model training procedure

j¼1

8 < subject to :

3

 ai  ai ¼ 0

(6)

0  ai ; ai  C; i ¼ 1; 2; /n

The following hyper-parameters of the SVM model should be properly determined for good generalization performance: the regularization factor (C), the width of the ε-intensive zone (ε), and the kernel coefficient (s). The regularization factor controls the trade-off between model complexity and training errors. The width of the ε-intensive zone determines the accuracy of fitting the training data and thus affects the model complexity. The kernel coefficient adjusts the width of the RBF kernel, which also affects the predictive performance of the derived model. In this study, a 10-fold cross-validation scheme was employed for setting those hyper-parameters [25].

The hybrid model has the following three groups of model parameters to be determined on the basis of the training data set: 1) the coefficients of the empirical equation (E0, b, R, p, and q), 2) the coefficients involved in the SVM model (a* and a), and 3) the hyper-parameters of the SVM (C, ε, and s). Fig. 2 describes the model training procedure for determining these

Data preparation Operational data were obtained from a full factorial design of experiments for a 1.2-kW PEM fuel-cell stack operating on high-pressure pure hydrogen and oxygen in a dead-end mode. The following five operating variables were selected as the input variables for the hybrid model: the current density (i), the hydrogen inlet pressure (PH2 ), the oxygen inlet pressure (PO2 ), the stack temperature (TS), and the oxygen relative humidity (RH). Three levels of value were assigned to each input variable except for the current density as listed in Table 1, and then 81 (¼34) tests were carried out under the conditions corresponding to all the possible combinations of the operating variables. At each combination, a polarization curve of the stack was measured while raising the current density from 0 to 1000 in 10e120 mA cm2 steps. At every increment of current density, the cell voltages were measured for 1 min and then averaged over both the time and the cells in the stack. These operational data were randomly divided into training and testing data sets in a 3:2 proportion (827 and 550 samples,

Table 1 e Operating variables and levels for the full factorial design of experiments conducted for obtaining the operational data. Operating variables Hydrogen inlet pressure (PH2 ) Oxygen inlet pressure (PO2 ) Stack temperature (TS) Oxygen relative humidity (RH)

Operating levels 150, 190, and 230 kPa 150, 190, and 230 kPa 45, 60, and 75  C 30%, 60%, and 100%

Fig. 2 e Model training procedure for determining the model coefficients and hyper-parameters of the hybrid model. R2 is defined by P b i Þ2 =Pn ðVi  VÞ2 where V is the mean R2 ¼ 1  ni¼1 ðVi  V i¼1 of the measured cell voltages.

Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131

4

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

parameters. In this study, the MATLAB programming that utilizes the global optimization toolbox [28] was carried out to implement the model training procedure. First, nonlinear least-square regression was performed using the LevenbergeMarquardt algorithm [29] to get the following estimates of the coefficients of the empirical equation: E0 ¼ 1.0723 V, b ¼ 0.0231 V, R ¼ 3.8176  104 kU cm2, p ¼ 0.0887 V, and q ¼ 1.2321  104 cm2 mA1. Next, the b emp ) from voltage differences (DV) between the predictions ( V the empirical equation and the measurements (V) were calculated, and then used as the target output vector in training the SVM model. The hyper-parameters of the SVM model were determined using a 10-fold cross-validation scheme, which proceeds in the following steps:  Step 1: Partition the training data set into 10 subsets.  Step 2: Select a single subset for model testing and the remaining nine subsets for model training.  Step 3: Obtain the current values of the hyper-parameters using the generalized pattern search algorithm [30].  Step 4: Perform model training with the current values of the hyper-parameters and then model testing to calculate the mean squared error (MSE) according to P b svm Þ2 . MSE ¼ n1 ni¼1 ðDVi  D V i  Step 5: Repeat Steps 2e4 10 times, with each of the 10 subsets chosen in turn for modeling.  Step 6: Compute the objective function by averaging the MSEs for the 10 subsets.  Step 7: Repeat Steps 2e6 until the objective function is minimized. The minimizer is the final values of the hyperparameters. The following optimal hyper-parameters were obtained at the minimum cross-validation error using the 10-fold crossvalidation scheme: C ¼ 291.0412, ε ¼ 0.0088, and s ¼ 0.1386. Finally, the model coefficients of the SVM model were obtained by training the model using both the entire training data set and the hyper-parameters.

Results and discussion The predictive performance of the hybrid model was compared with that of a straight SVM model which was built b The optimal hyperto directly predict the cell voltage ( V). parameters (C ¼ 268.1921, ε ¼ 0.0226, and s ¼ 0.1486) of the SVM model were found using the 10-fold cross-validation scheme as used for the hybrid model, except that P b i Þ2 was applied in Step 4. First, both the MSE ¼ n1 ni¼1 ðVi  V hybrid and SVM models were constructed using the training data set, and then were evaluated using the testing data set to generate predicted cell voltages. Fig. 3 plots the predicted cell voltages using the hybrid model (Fig. 3a) and the SVM model (Fig. 3b) against the measured cell voltages for the testing data set. Comparing the two figures clearly indicates the superiority of the hybrid model to the SVM model: the cell voltages predicted by the former excellently match the measured voltages with most data points falling on a line of slope 1, whereas those predicted by the latter are scattered along the line with noticeable deviations in the open circuit voltage

Fig. 3 e Comparison of the predicted cell voltages with the measured ones in the testing data set: (a) cell voltages predicted using the hybrid model and (b) cell voltages predicted using the SVM model.

region. The superiority of the hybrid model is quantitatively confirmed again by its higher R2 (coefficient of determination) value of 0.9996 than its counterpart of 0.9844. Considerable deviations between predicted and measured cell voltages in the open circuit voltage regions were also reported by some previous studies on SVM modeling for polarization curve prediction [24,25]. Similar deviations were reproduced by the straight SVM model in this study as can be seen in Fig. 3b, even though the model was constructed using a data set obtained from a full factorial design of experiments which used quite small incremental steps (10 mA cm2) of the current density in the high voltage region. An additional comparison was made between the hybrid and SVM models using the validation data set obtained at the two different operating conditions: OP1 and OP2. Fig. 4 shows the polarization curves fitted using both the models along with the measured data points and absolute prediction errors. The hybrid model shows almost perfect matches between the predicted and measured polarization curves for both the operating conditions, whereas the SVM model exhibits

Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

5

Fig. 4 e Comparison of the predicted polarization curves and absolute prediction errors of the hybrid model with those of the SVM model, along with the measured data points at the two different operating conditions: (a) OP1 and (b) OP2.

Table 2 e Comparison of the predictive performance between the hybrid and SVM models for the validation data set. Operating conditions

Performance Hybrid SVM criteria model model

R2 OP1: PH2 ¼ 230 kPa, PO2 ¼ 230 kPa, RMSEa TS ¼ 70  C, and RH ¼ 80% MaxErrorb R2 OP2: PH2 ¼ 150 kPa, PO2 ¼ 150 kPa, RMSEa  TS ¼ 50 C, and RH ¼ 40% MaxErrorb qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi P n 2 a b RMSE ¼ n1 i¼1 ðV  i  V i Þ .  b b i MaxError ¼ maxVi  V  for i ¼ 1; 2; /; n.

0.9999 1.7 mV 3.1 mV 0.9999 2.2 mV 4.4 mV

0.9805 12.3 mV 34.8 mV 0.9892 10.6 mV 39.6 mV

relatively poor matches. The predictive performance of the two models is clearly distinguishable in the plots of the absolute prediction errors, which exhibit much lower errors for the hybrid model than for the SVM model across the entire current density range for both the operating conditions. The quantitative performance criteria listed in Table 2 also confirm the superiority of the hybrid model to the SVM model. The hybrid model showed extremely high R2 values close to 1 (0.9999 for both OP1 and OP2) and significantly lower RMSEs (root-mean-square errors) (1.7 and 2.2 mV for OP1 and OP2, respectively), which correspond to only 14 and 21% of the RMSEs obtained from the SVM model.

Conclusion A hybrid model that combines a SVM model with a simple empirical equation was proposed for more accurate prediction of the polarization curves of a PEM fuel cell as a function of various operating variables. Operational data were obtained from designed experiments for the PEM fuel cell for training, testing, and validating the model. A model training procedure

was also presented for determining the model coefficients and hyper-parameters of the hybrid model on the basis of the training data set. The polarization curves predicted by the trained hybrid model almost perfectly matched the measured ones for both the testing and validation data set. The hybrid model successfully compensated for the drawback of the SVM model that generated large prediction errors in the high voltage range of the polarization curves. The proposed hybrid model has a simple structure consisting of a single empirical equation with five coefficients and a SVM model and can be used to predict the cell voltages of PEM fuel cells with high accuracy even when the major operating variables, namely, the current density, the stack temperature, the hydrogen and oxygen inlet temperatures, and the oxygen relative humidity, vary.

Appendix A. Supplementary data Supplementary data related to this article can be found at http://dx.doi.org/10.1016/j.ijhydene.2017.01.131.

references

[1] Wang C-Y. Fundamental models for fuel cell engineering. Chem Rev 2004;104:4727e66. [2] Weber AZ, Borup RL, Darling RM, Das PK, Dursch TJ, Gu W, et al. A critical review of modeling transport phenomena in polymer-electrolyte fuel cells. J Electrochem Soc 2014;161:F1254e99. [3] Shah AA, Luo KH, Ralph TR, Walsh FC. Recent trends and developments in polymer electrolyte membrane fuel cell modeling. Electrochim Acta 2011;56:3731e57. [4] Napoli G, Ferraro M, Sergi F, Brunaccini G, Antonucci V. Data driven models for a PEM fuel cell stack performance prediction. Int J Hydrogen Energy 2013;38:11628e38.

Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131

6

i n t e r n a t i o n a l j o u r n a l o f h y d r o g e n e n e r g y x x x ( 2 0 1 7 ) 1 e6

[5] Petrone R, Zheng Z, Hissel D, Pera MC, Pianese C, Sorrentino M, et al. A review on model-based diagnosis methodologies for PEMFCs. Int J Hydrogen Energy 2013;38:7077e91. [6] Cheng S-J, Liu J-J. Nonlinear modeling and identification of proton exchange membrane fuel cell (PEMFC). Int J Hydrogen Energy 2015;40:9452e61. [7] Li Q, Chen W, Liu Z, Guo A, Huang J. Nonlinear multivariable modeling of locomotive proton exchange membrane fuel cell system. Int J Hydrogen Energy 2014;39:13777e86. [8] Curteanu S, Piuleac C-G, Linares JJ, Canizares P, Rodrigo MA, Lobato J. Neuro-evolutionary approach applied for optimizing the PEMFC performance. Int J Hydrogen Energy 2014;39:4037e43. [9] Kheirandish A, Shafiabady N, Dahari M, Kazemi MS, Isa D. Modeling of commercial proton exchange membrane fuel cell using support vector machine. Int J Hydrogen Energy 2016;41:11351e8. [10] Salva JA, Iranzo A, Rosa F, Tapia E. Experimental validation of the polarization curve and the temperature distribution in a PEMFC stack using a one dimensional analytical model. Int J Hydrogen Energy 2016;41:20615e32. [11] Wang Y-X, Ou K, Kim Y-B. Modeling and experimental validation of hybrid proton exchange membrane fuel-cell/ battery system for power management control. Int J Hydrogen Energy 2015;40:11713e21. [12] Salva JA, Iranzo A, Rosa F, Tapia E, Lopez E, Isorna F. Optimization of a PEM fuel cell operating conditions: obtaining the maximum performance polarization curve. Int J Hydrogen Energy 2016;41:19713e23. [13] Moreira MV, da Silva GE. A practical model for evaluating the performance of proton exchange membrane fuel cells. Renew Energy 2009;34:1734e41. [14] Beicha A. Modeling and simulation of proton exchange membrane fuel cell systems. J Power Sources 2012;205:335e9. [15] Haji S. Analytical modeling of PEM fuel cell i-V curve. Renew Energy 2011;36:451e8. [16] Kim J, Lee S-M, Srinivasan S, Chamberlin CE. Modeling of proton exchange membrane fuel cell performance with an empirical equation. J Electrochem Soc 1995;142:2670e4.

[17] Saadi A, Becherif M, Aboubou A, Ayad MY. Comparison of proton exchange membrane fuel cell static models. Renew Energy 2013;56:67e71. [18] Laurencelle F, Chahine R, Hamelin J, Agbossou K, Fournier M, Bose TK, et al. Characterization of a Ballard MK5-E proton exchange membrane fuel cell stack. Fuel Cells 2001;1:66e71. [19] Ou S, Achenie LEK. A hybrid model neural network model for PEM fuel cells. J Power Sources 2005;140:319e30. [20] Zhong Z-D, Zhu X-J, Cao G-Y, Shi J-H. A hybrid multi-variable experimental model for a PEMFC. J Power Sources 2007;164:746e51. [21] Lee W-Y, Park G-G, Yang T-H, Yoon Y-G, Kim C-S. Empirical modeling of polymer electrolyte membrane fuel cell performance using artificial neural networks. Int J Hydrogen Energy 2004;29:961e6. [22] Han I-S, Shin HK. Modeling of a PEM fuel cell stack using partial least squares and artificial neural networks. Korean Chem Eng Res 2015;53:236e42. [23] Lobato J, Canizares P, Rodrigo MA, Linares JJ, Piuleac C-G, Curteanu S. The neural networks based modeling of a polybenzimidazole-based polymer electrolyte membrane fuel cell: effect of temperature. J Power Sources 2009;192:190e4. [24] Zhong Z-D, Zhu Z-J, Cao G-Y. Modeling a PEMFC by a support vector machine. J Power Sources 2006;160:293e8. [25] Han I-S, Chung C-B. Performance prediction and analysis of a PEM fuel cell operating on pure oxygen using data-driven models: a comparison of artificial neural network and support vector machine. Int J Hydrogen Energy 2016;41:10202e11. [26] Vapnik V. The nature of statistical learning theory. 2nd ed. New York: Springer; 1999. [27] Huo H-B, Zhu X-J, Cao G-Y. Nonlinear modeling of a SOFC stack based on a least squares support vector machine. J Power Sources 2006;162:1220e5. [28] Mathworks. Global optimization toolbox user's guide. 2011. [29] Rao SS. Engineering optimization: theory and practice. 4th ed. Hoboken, New Jersey: John Wiley & Sons; 2009. [30] Kolda TG, Lewis RM, Torczon V. Optimization by direct search: new perspectives on some classical and modern methods. SIAM Rev 2003;45:385e482.

Please cite this article in press as: Han I-S, Chung C-B, A hybrid model combining a support vector machine with an empirical equation for predicting polarization curves of PEM fuel cells, International Journal of Hydrogen Energy (2017), http://dx.doi.org/10.1016/ j.ijhydene.2017.01.131