Non-linear Predictive Control with Multi Design Variables for PEM-FC

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1 Non-linear Predictive Control with Multi Design Variables for PEM-FC A. Shokuhi-Rad, M. Naghash-Zadegan, N. Nariman-Zadeh, A. Jamali, A.Hajilu Abstract Designing of a non-linear controller base on model predictive control for Proton Exchange Membrane Fuel Cells (PEMFC) was presented by M. Pucci et al. to regulate the voltage based on the hydrogen pressure, trying to reduce the variation of the input control variable [1], but they used a constant signal for other important control variables like operating temperature and current density duo to computational limitations in online optimization of multivariable highly non-linear system. In this work, by use of Approximate Predictive Control (APC) method based on neural network model, the whole control process is designed with three input variables. We assume operating temperature and hydrogen pressure as control design variable and current density as measured disturbance to manage the voltage. A comparison with various design condition is presented too. Keywords Approximate Predictive Control, Neural Networks, Neural Networks, PEMFC. T I. INTRODUCTION HE need for reliable power source is an inseparable part of human life. This seems more necessary in under development countries as the main infrastructure. In many countries this need is neglected due to existence of fossil fuel sources but the fact that these sources are limited is undeniable. It should be noted that the dependence of a country in terms of energy supply can lead to other sequential dependence on the owners. Also with the current environmental problems and air pollution, the need for public efforts to reach clean sources of energy is urgent. Therefore, nowadays access to clean and renewable sources of energy is vital. In this respect, the Fuel technology has attracted much attention because of inherent properties and potentials in its technology. Although this technology is under development for more than a decade, optimizing the efficiency and reducing costs are still in progress. In this work, the Model Predictive Control (MPC) is applied to control the output voltage of a PEM fuel stack A.Shokuhi-Rad is with the Department of Mechanical Engineering, Guilan University, Iran, Rasht, PO Box (Corresponding author to provide phone: ; sa.rade@alumni.com) M.Naghashzadegan is with the Department of Mechanical Engineering, Guilan University, Iran, Rasht, PO Box ( naghash@guilan.ac.ir) N.Nariman-Zadeh is with the Department of Mechanical Engineering, Guilan University, Iran, Rasht, PO Box ( nnzadeh@guilan.ac.ir) A.Jamali is with the Department of Mechanical Engineering, Guilan University, Iran, Rasht, PO Box ( ali.jamali@guilan.ac.ir) A.Hajiloo is with the Department of Mechanical Engineering, Guilan University, Iran, Rasht, PO Box ( ahajiloo@gmail.com) consist of 35 single s. The use of operating temperature and input pressure as the two main design variables have made distinctive of previous attempts. To achieve this goal, a non-linear neural network model with a complete feed forward structure is proposed. Then, based on this model, an MPC controller is employed to approach the desired output voltage. However the main challenge here is to accomplish the optimization process of a highly non-linear model with the two design variables. II. ELECTROCHEMICAL MODELING OF PEMFC The PEM-FC converts chemical energy into electric one, by employing hydrogen (H ) as fuel and oxygen (O ) as oxidizer, giving heat and water as undesired products. The development of a generalized steady-state electrochemical model of a Proton Exchange Membrane Fuel Cell (PEMFC) system is presented by R.F Mann et al. to predict the output voltage of a single Ballard Mark V as a function of the major variables like current density, pressure of the reactant gases and working temperature more efficiently []. But this model cannot explain the behavior of the voltage at high current densities as well as the other ranges. J. M. Corrêa et al. took another term for mass transfer losses into account; this model can predict output voltage in all ranges [3]. Using this model, the power density of a Ballard Mark V fuel system that consists of 35 s is predicted as a function of current density, operating temperature and hydrogen pressure. The output voltage of a single can be defined as the result of the following expression [1], [4] V = E + V + V + V (1) Nernst act ohm Where: E Nernst is the thermodynamic potential of the, representing its reversible voltage; V act is the voltage drop due to the activation of the anode and cathode; V ohm is the ohmic voltage drop; V con represents the voltage drop resulting from the reduction in concentration of the reactants gases or, alternatively, from the transport of mass of oxygen and hydrogen. The total voltage output of the fuel unit with 35 s connected in series is [4] V = n V () stack con 198

2 A. Cell reversible voltage The reversible voltage of the (E Nernst ) is the potential of the obtained in an open circuit thermodynamic balance (without load). In this model, E Nernst is calculated starting from a modified version of the equation of Nernst [3] E Nernst G S = + ( T Tref ) F F RT + F [ ln( P ) + 1 ln( P )] H O B. Activation Voltage Drop The total expression to represent the activation overpotential, including anode and cathode, can be calculated by V act (3) = ξ 1 + ξt + ξ3t ln( C ) + ξ4t ln( JA) (4) O Where the ξ s represent parametric coefficients for each model and their values are defined based on theoretical equations with kinetic, thermodynamic and electrochemical foundations [3]. C. Ohmic Voltage Drop The ohmic voltage drop results from the resistance to the electrons transfer through the collecting plates and carbon electrodes, and the resistance to the protons transfer through the solid membrane. In this model, a general expression for resistance is defined to include all the important parameters of the membrane, given by. [1], [3] accumulation on their surfaces or a load transfer from one to another occurs. The charge layer in correspondence of the electrolyte/electrode interface behaves as a storage of electrical charges, and therefore, from the electric circuit point of view, can be represented by a capacitor. At each voltage variation, a time is required for charging, in case of voltage increase, or vanishing, in case of voltage decrease. This time delay affects the activation and concentration overpotential, and not the ohmic drop, whose variation can be, however, considered instantaneous. The activation and concentration overpotentials can be modeled as first-order delay elements with a time constant τ = CR a (7) Where C is the equivalent capacitance in [F] and Ra is the equivalent resistance in [Ω]. The time constant τ governing the dynamics is variable with the load conditions, since the equivalent resistance Ra is a function of the activation and concentration overpotentials and load current [1], [4]. F. Simulation Result For validation of the model, one single, model Ballard Mark V, was simulated, which was fed with gases H and O, using the membrane Nafion 117. The parameters used for this simulation are presented in table (1) [3] By use of these parameters and proposed equations, the polarization curve and power density of Ballard Mark V PEM fuel is illustrated in Figure (1) respectively. V ohm ( prot + Relec ) = JARint ernal = (5) JA R Where R prot represents equivalent resistance of the membrane. D. Concentration Voltage Drop To determine an equation for this voltage drop, a maximum current density was defined J max, under which the fuel was being used at the same rate of the maximum supply speed. The current density cannot surpass this limit because the fuel cannot be supplied at a larger rate given by [3] J V con = B ln 1 (6) J max Where B is a constant that depends on the fuel and its operating state (V). E. Dynamics of the Cell The dynamics of the is mainly governed by the so called charge double layer effect. When two differently charged materials are kept in contact, either a charge Symbol TABLE I BALLARD MARK V SINGLE CELL PARAMETERS Value A 50.6 cm l 178 μm p H 1 atm p O 1 atm B V R elec ohm ξ ξ lnA+(4.3x10-5).ln(C H ) ξ ξ ψ 3 J max 1500 ma/cm 199

3 IV. PREDICTIVE CONTROL DESIGN For the Nonlinear Predictive Control (NPC) the predictor is given by the successive recursion of a deterministic neural network model. The predictor is nonlinear in the future control inputs. The optimization problem must be solved at each sample, resulting in a sequence of future control inputs u(t). From this sequence, the first component u(t) is then applied to the system. The minimization criterion is as follow [6] J ( N, N, N ) 1 u N = j= N1 y + λ ( t + j t) w( t + j) N u j= 0 [ u( t + j) u( t + j 1) ] (8) Fig. 1 Ballard Mark V Polarization Curve III. NEURAL MODELING OF PEMFC This work proposes a feedforward neural model design, in which the system is modeled (identified) by a neural approach. The neural approach may consist of an ensemble of neural networks which identify different operating ranges. Each neural network is a Multilayer Perceptron (MLP), which is a supervised feedforward network. Its adjustable parameters are called weights and are determined from a set of examples (training set, TS) through the process called training. Weights are computed by the prediction error approach, based on the minimization of a measure of closeness in terms of a sum-of-squares error criterion. This minimization is achieved by an iterative search scheme called Levenberg-Marquardt, which is a trust-region method based on a second-order approximation of the criterion around the current iterate (Gauss-Newton method). The choice of the inputs (regressor vector) for each neural network is very important. Here a Neural Network AutoRegressive external input (NNARX) model structure has been chosen: the regression vector is composed of the past n inputs and m outputs of the MLP. NNARX is always stable even if the system is unstable, because there is a pure algebraic relationship between prediction and past measurements and inputs [1], [5]. Figure () presents a schematic of an NNARX system: Where w is the reference signal, N 1 the minimum prediction horizon (here set equal to d), N the prediction horizon, N u the control horizon, λ the weighting factor penalizing changes in the control input. y ( t + j t) represents the minimum variance k-step head predictor. The NPC strategy may have several local minima and is computationally demanding. In order to speed up the method, it can be replaced by the Approximate Predictive Control (APC) which applies the instantaneous linearization principle: at each sample a linear model is extracted from a neural network model of the system and a linear controller is designed. Obviously this can be valid only around the operating point [1]. In APC the predictor is given by an approximate minimum variance estimator based on instantaneous linearization of a NNARX model (here an integrated ARX model, ARIX, is achieved). There exists a unique solution of criterion and the future control inputs can be found directly. Hence, it is faster than NPC, but may have a limited validity in certain regimes of the operating range [1]. Figure (3) represents a schematic of APC control design of PEMFC Fig. 3 A schematic of APC control design Fig. A schematic of an NNARX model V. SIMULATION RESULTS Base on dynamic model of PEMFC, 5000 pairs of inputoutput data has been generated and used for training and 00

4 validating the neural model. We randomly selected 60 percent of data for training and 40 percent for validating the model. For each training set a MLP has been trained for obtaining the forward model. The architecture of the neural network consists of one hidden layer of 9 neurons with mean square error as activation function, and one linear output neuron. The regression vector (NNARX), as explained above, is made up of the voltages and pressures of the two previous time samples; the time delay is equal to one time sample. The training has been accomplished by the Levenberg- Marquardt method. To validate the estimated model a test set has been created and the tests for correlation with different combinations of past residuals (prediction errors) and data have been performed. Figure (4) represents predicted output of neural model versus real output. The resemblance between the actual data and this model in more than 99%. Fig. 6 Current density as a measured disturbance VI. MULTI-OBJECTIVE OPTIMIZATION Here we use MUGA which is a modified version of NSGA II to achieve the optimum designs base on three objective functions. These three objectives are respectively: square of error between controlled output and desired one, square of deviation of pressure from lower allowed boundary, square of deviation of temperature from lower allowed boundary and the respective design variables are: prediction horizon, control horizon, and two weight associated to each inputs. Fig. 4 Actual Output vs. Modeled Output To design the controller, the desire output and current density as measured disturbance has been presumed as below (figures (5), (6)) Objective (min of ) sq. of Temperature sq. of Pressure sq. of Error TABLE II DESIGNS PARAMETERS ASSOSIATED TO EACH CASE Prediction Horizon Control Horizon Weight associated to P Weight associated to T Based on the parameters presented in Table (), the predictive controller is designed four times by replacing the optimum value of prediction horizon, control horizon and the weights associated to inputs. Their corresponding control variables and outputs are presented in figures as follow: Fig. 5 Desired output of the stack 01

5 design mentioned as square of Error offers a good performance on desired output tracking but poor performance on variation of input variables. As it is obvious in the above figures, the forth design mentioned as Normalized Distance offers the best trade-off between other design cases due to a fast response and lower input variations. It also gets advantages of lower computational power because in each step, the optimization problem solves for only two steps forward but still shows a good performance either in variation of input variables and desired output tracking. Fig. 7 Controlled output vs. desired output REFERENCES [1] M. Cirrincione, M. Pucci, G. Cirrincione, M. G. Simões, A Neural Nonlinear Predictive Control for PEM-FC, Journal of electrical system 1- (005): [] R. F. Mann; J. C. Amphlett; M. A. I. Hooper; H.M. Jensen; B. A. Peppley and P. R. Roberge; Development and application of a generalised steady-state electrochemical model for a PEM fuel ; Journal of Power Sources 86; 000; pp [3] J. M. Corrêa, F. A. Farret, L. N. Canha, and Marcelo G. Simões, An Electrochemical-Based Fuel-Cell Model Suitable, for Electrical Engineering Automation Approach, IEEE Trans. Ind., Electr., VOL. 51, NO. 5, October 004. [4] J. Larminie, A. Dicks, Fuel Cell Systems Explained (Second Edition), John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England, 003. [5] Nand Kishor, S. P. Singh, "Nonlinear predictive control for a NNARX hydro plant model", Journal of Neural Computing & Applications, pp , Springer London, 006. [6] E.F. Camacho, C. Bordons, Model Predictive Control, Springer, 004. Fig. 8 Hydrogen pressure as a design variable Fig. 9 Operating temperature as a design variable VII. CONCLUSION The first and second designs mentioned as square of Temperature and square of Pressure offer a good performance on variation of input variables but poor performance on desired output tracking vice versa the third 0