J. Cent. South Univ. Technol. (2007)05-0690-06

DOI: 10.1007/s11771-007-0132-y                                                                                      

Estimation of equivalent internal-resistance of PEM fuel cell

using artificial neural networks

LI Wei(李  炜), ZHU Xin-jian(朱新坚), MO Zhi-jun(莫志军)

(Department of Automation, Fuel Cell Research Institute, Shanghai Jiaotong University, Shanghai 200030, China)

Abstract：A practical method of estimation for the internal-resistance of polymer electrolyte membrane fuel cell (PEMFC) stack was adopted based on radial basis function (RBF) neural networks. In the training process, k-means clustering algorithm was applied to select the network centers of the input training data. Furthermore, an equivalent electrical-circuit model with this internal-resistance was developed for investigation on the stack. Finally using the neural networks model of the equivalent resistance in the PEMFC stack, the simulation results of the estimation of equivalent internal-resistance of PEMFC were presented. The results show that this electrical PEMFC model is effective and is suitable for the study of control scheme, fault detection and the engineering analysis of electrical circuits.

Key words: polymer electrolyte membrane fuel cell(PEMFC); equivalent internal-resistance; radial basis function; neural networks     

1 Introduction

Fuel cells have become one of the major areas of research in academia and the industry with the numerous advantages better than the batteries and especially than the other small-scale sources of electricity, including the photovoltaic and solar cells^[1]. The main factors to increase the interest in PEMFC are as follows: 1) low operating temperature that permits short start-up and shut-down periods and rapid response to load variations; 2) low pressure that allows safety; 3) the by-product waste is water; 4) low emission and high efficiency; 5) modularity; 6) potential cost reduction due to a scale economy.

It is necessary to use an estimated method for an equivalent resistance of an electrical PEMFC model. It is difficult to model the PEMFC theoretically based on mechanistic approaches because various crucial variables should be considered in the modeling procedure. Though assumptions and approximations have to be made for the purpose of developing the mechanistic models, these PEMFC models are still complex and hard to be operated.

In order to avoid having to acquire a good knowledge of physicochemical phenomena and the process parameters, “black-box” models are used to achieve behavioral modeling of PEMFC^[2]. These models just make mappings for the relationships between the inputs and outputs with the experimental data, and do not reflect the inner states of PEMFC during the operation. Although the electrical PEMFC model described in this paper is established through RBF neural networks and belongs to “black-box” model, it is not complex and the parameter (nonlinear internal resistance) has an explicit mechanistic explanation that is also helpful to analyze the actual PEMFC inner-performance.

In this paper, a brief description of the physical structure and the operating principles of a PEMFC were presented. Moreover, a set of equations was used to represent an electrical PEMFC model and the definition of the internal-resistance was described in detail. The framework of RBF neural networks and the algorithm for the parameter compensation of the internal-resistance were illustrated and the on-line electrical PEMFC model was established.

2 Physical structure and electrical model of PEMFC

A typical PEMFC is shown in Fig.1. It consists of an anode and a cathode electrode with a proton-conducting membrane as the electrolyte sandwiched in between^[3]. H₂ obtained from methanol (CH₃OH), petroleum products or natural gas acts as fuel. It is fed through a narrow channel from one end of the plate (Anode, negative electrode). Similarly, O₂ enters the fuel cell from the other end of the plate (Cathode, positive electrode).

Fig.1 Schematic diagram of PEMFC

H₂ and O₂ have a strong chemical affinity and hence the membrane separating the two gases allows only H⁺ to pass through it. While H⁺ takes the shortest path, i.e. through the membrane, electrons travel all the way around the external circuit to create useful current. H⁺ ions combine with oxygen to produce water and heat energy as by-products in this case.

The PEMFC can operate on air. The performance of fuel cell can be improved by pressurizing the air. In any application, there will be a trade-off between the energy and financial cost associated with the higher pressures air and the improved performance. Because the PEMFC operates at low temperatures and does not contain a liquid electrolyte, catalyst migration and re- crystallization are not consideration.

The chemical reactions occurring at the oxidation and reduction electrode of PEMFC are as follows:

2H₂→4H⁺+4e^-                 (1)

O₂+4H⁺+4e^-→2H₂O              (2)

The total cell reaction is

O₂+2H₂→2H₂O                (3)

A number of approaches have been used to model the PEMFC behavior^[3]. The parameter model of PEMFC developed by AMPHLETT et al^[4-5] used a mechanistic approach with a number of parameters. The fuel cell voltage V is defined as the three terms: the thermodynamic potential minus, the activation over potential and ohmic over potential. The ideal standard potential (E₀) of a PEMFC (a H₂/O₂ fuel cell) is 1.229 V with liquid water product. The actual cell potential  decreases from its equilibrium potential because of the irreversible losses. Several sources of polarization over-potential are contributed to the irreversible losses in a practical fuel cell, namely activation polarization, ohmic polarization and concentration polarization. These losses result in a cell voltage V for a fuel cell that is lower than its ideal potential E.

The equivalent electrical model of a single PEMFC is indicated in Fig.2. The resistance R_int models the ohmic losses. The activation resistance R_a models the activation over potential. The effect of double charge layer is modeled by a capacitor C_d connected in parallel with R_a, and the dynamics of fuel cell voltage can also be modeled by an addition of C_d, which “smoothes” any voltage drop change across this R_a. If the concentration over potential is considered, it would be incorporated into this resistance. Generally speaking, the effect of this capacitance resulting from the charge double layer gives the fuel cell a “good” dynamic performance, in that the voltage moves gently and smoothly to a new value in response to a change in current demand^[6].

Fig.2 Equivalent electrical model of a fuel cell

Assume that the fuel cell system consists of a stack of n similar cells connected in series, therefore the total stack voltage is given by

                                   (4)

Because the voltage loss of the open-circuit polarization V_o,pol exists when the demand current is zero, thus the open-circuit voltage can be written as follows:

V_o=n (E-V_o,pol)                (5)

In the steady state, 

              (6)

where  V_o is the open-circuit stack voltage,  is the stack equivalent internal-resistance, i is the stack current. An electrical steady-state model of PEMFC stack is illustrated in Fig.3.

Fig.3 Electrical steady-state model of a PEMFC stack

 The consumed power of the stack equivalent resistance  is written by   

                   (7)

3 Estimation of equivalent resistance for PEMFC model

3.1 RBF neural networks

A RBF neural network consists of an input layer, a nonlinear hidden layer and a linear output layer. The nodes within each layer are fully connected to the previous layer nodes. The input variables are each assigned to nodes in the input layer and connected directly to the hidden layer without weights. The hidden layer nodes are RBF units that calculate the Euclidean distances between the centers and the network input vector, and output the results through a nonlinear function^[7]. The output layer nodes are weighted linear combinations of the RBF in hidden layer. Fig.4 shows the structure of a RBF neural network with m inputs, p outputs and N hidden nodes.

Fig.4 Structure of RBF neural network

The hidden unit can be expressed as a matrix , the neural networks weight  the input vector  the output vector   is a nonlinear function and is chosen as a Gaussian activation function:

      (8)

where  is the center of the jth hidden unit, and with the same dimension as the input vector X(k), λ_j is the width of the jth hidden unit, ||?|| means the Euclidean norm.

Then the ith RBF network output can be represented as a linearly weighted sum of N basis functions:

    (9)

where w_j,i and w_0,i are the weights, w_0,i is used to compensate for the difference between the average value over the data set of the RBF activation and the corresponding average value of the target outputs.

From Fig.4, the transformation from the input layer to the hidden layer is nonlinear, because of using Gaussian functions Ψ(?) as RBFs.

3.2      Estimation of equivalent resistance with RBFNN

Many parameters, such as pressure, temperature, gas flow, humidity level and so on, affect the equivalent resistance. For the training and simulation process of the RBFNN, data on different equivalent-resistance and current characteristics of the fuel cell are used.

         The RBFNN model has 10 input nodes and a linear output neuron for estimating the equivalent-resistance  of a fuel cell stack. Through the estimated equivalent- resistance, the internal variation of the fuel cell stack related can be estimated.

The general scheme of modeling the PEMFC stack with RBFNN is shown in Fig.5. It includes off-line estimation of the equivalent internal- resistancewith RBFNN, on-line compensation of the equivalent internal-resistance, and on-line modeling for the PEMFC stack with the compensated internal-resistance.

Fig.5 Schematic diagram of on-line modeling for PEMFC stack with RBFNN

3.2.1 Off-line estimation of equivalent internal- resistance with RBFNN

Using the RBFNN, the equivalent resistance  of stack against current and process variables can be described by the following function:

   (10)

where  i is the stack current; t_s is the stack temperature; p_a, p_c are the inlet anode pressure and cathode pressure, respectively; t_a,w, t_c,w are the temperatures of the humidifying water vapor; H_a, H_c are the relative humidity respectively for the anode and cathode; and are the stoichiometry factors of air and hydrogen.

In this case, the input vector of RBFNN x₇(k), x₈(k), where m=1,…,10 correspond to the sequence in Eqn.(10), and the output vector . To train RBFNN, normalization is necessary to transform all the input and output data into the range of (0,1) as follows:

          (11)

where  D is the original data, D_max is the maximum of D, D_min is the minimum of D, D′ is the result of normalization, 0＜d₁＜d₂＜0.1. Here, d₁=0.05, d₂=0.1 are chosen.

The training of RBFNN is to minimize the following quadratic cost function of the output error between the output values of the network and the experimental measurements:

          (12)

where  M is the number of patterns.

A common training of the RBF neural networks can be divided into two stages procedures^[8]. The first stage is to choose the network center vectors C_j that reflect the probability density of the input data by using an unsupervised technique^[7]. Here k-means clustering algorithm is applied to select the network centers of the input training data^{[7, 9]}. After selecting the network centers, the output layer weights are determined in the second training stage using a recursive least squares algorithm (RLSA)^[10].

                                           (13)

where  W₁(k) is the weight vector of the output layer when sample k is used for training and ψ(k) is the output vector of the hidden layer; L₁(k) is the gain vector; initial P₁(0)=μI, μ is a huge positive number, and μ＞10⁵. The start points of W₁(k) pick up the weights of the output layer arbitrarily; β₁ is the forgetting factor; W₁(k+1) is the updated weight vector of the corresponding network output, and thus it will cause  to be updated as well. The width parameters λ_j of the RBF may be calculated by many techniques. Here all widths are equal to about twice the average space between the basis function centers^[7].

3.2.2 On-line compensation of equivalent internal- resistance

After the off-line training of RBFNN, an off-line model of PEMFC is developed for the equivalent internal-resistance In order to implement the

dynamic real-time correction, a technique of on-line compensation for off-line PEMFC model is developed as shown in Fig.5.

 At k=1, 2, …, S operation states, calculate the resistance value of compensation ΔR(K), is the sampled output voltage of stack, is the output voltage of the off-line PEMFC model with the equivalent resistance , is the sampled output current of stack ().

        (14)

and then the compensated total inter-resistance of the off-line PEMFC can be expressed as

                                                  (15)

After parameter compensation storage, the on-line PEMFC model can be written as

,       (16)

where is the compensated output voltage of PEMFC model.

4 Simulation results and discussion

In order to perform the estimation of the equivalent resistance of the PEMFC stack with RBFNN, a set of experimental data are used for the training procedure. The experimental data are obtained from the PEMFC stack that consists of 24 single fuel cells connected in series. The operating condition parameters and physics characteristic parameters of PEMFC stack are listed in Table 1. More than 1 000 groups of experimental data are collected for training under different temperatures and gas pressures. The equivalent resistance values calculated by the neural network are compared with actual sampled resistance data obtained from the experiments of PEMFC stack.

The experimental results of estimation of with RBFNN are shown in Fig.6. The off-line estimation error is 0.2% and the estimation of  is accurate. The RBFNN can trace the variation of the internal-resistance well. Since the internal-resistance and its consumed power can be estimated, some internal states of PEMFC can be obtained. The validation results of the estimation of with RBFNN are shown in Fig.7. It can be seen from Fig.7 that the voltage output of electrical PEMFC model with  can also follow that of actual PEMFC stack well. The results show that this simple model can be applied to predict the PEMFC stack voltage as a function of the stack current and process variables over the whole current range.

Table 1 Experimental parameters of PEMFC stack

Fig.6 Experimental results and estimation of  with RBFNN

1—t_s=65 ℃, p_a=0.15  MPa, p_c=0.15 MPa; 2—t_s=70 ℃, p_a=0.15 MPa, p_c=0.15 MPa; 3—t_s=70 ℃, p_a=0.18 MPa, p_c=0.18 MPa; 4—t_s=75 ℃, p_a=0.18 MPa, p_c=0.20 MPa;     5—t_s=70 ℃, p_a=0.15 MPa, p_c=0.15 MPa

Fig.7 Output voltage of electrical PEMFC model and actual PEMFC stack (training)

 1—t_s=65 ℃, p_a=0.15 MPa, p_c=0.15 MPa; 2—t_s=70 ℃, p_a=0.15 MPa, p_c=0.15 MPa; 3—t_s=70 ℃, p_a=0.18 MPa, p_c=0.18 MPa; 4—t_s=75 ℃, p_a=0.18 MPa, p_c=0.20 MPa;  5—t_s=70 ℃, p_a=0.15 MPa, p_c=0.15 MPa  

Validation results are shown in Figs.8 and 9. It can be seen that the proposed correlation predicted the experimental data well. The errors of 92.5% experimental data versus predicted data are less than 15%. In fact, although the results of RBFNN training are good, there are some errors between the model output and the experimental data in the RBFNN validation. All these errors can be corrected with the on-line compensation scheme described before. After that the off-line model can be applied to on-line operation.

Fig.8 Experimental data and validation results of estimation of  with RBFNN

1—t_s=65 ℃, p_a=0.12 MPa, p_c=0.15 MPa; 2—t_s=75 ℃, p_a=0.18 MPa, p_c=0.18 MPa; 3—t_s=80 ℃, p_a=0.18 MPa, p_c=0.18 MPa

Fig.9 Output voltage of actual PEMFC stack and validation results of electrical PEMFC model

1—t_s=65 ℃, p_a=0.12 MPa, p_c=0.15 MPa; 2—t_s=75 ℃, p_a=0.18 MPa, p_c=0.18 MPa; 3—t_s=80 ℃, p_a=0.18 MPa, p_c=0.18 MPa

5 Conclusions

1) The development of PEMFC dynamic model with the estimated internal resistance can avoid complex calculations and derivations of formulas and is not like a “black-box” model whose parameters have no mechanistic explanations.

2) The estimation for the equivalent internal- resistance with RBFNN is effective and operational. The estimation of the internal-resistance with RBFNN has the ability to knowledge generalization that is good to understand the performances and the operational variation of PEMFC stack.

3) This model is simple and of on-line self-tuning. It can be applied to the study of designing real-time controller and fault detection of PEMFC stack. All kinds of fault data on flooding, drying and catalyst poisoning that affect the internal resistance are measured.

References

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Foundation item: Project (2003AA517020) supported by the National High-Tech Research and Development Program of China

Received date: 2006-10-20; Accepted date: 2006-12-23

Corresponding author: LI Wei, PhD; Tel: +86-21-34201547; E-mail: li_wei@sjtu.edu.cn

 (Edited by ZHAO Jun)