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This paper presents a nonlinear control strategy utilizing the linearization and input-output decoupling approach for a nonlinear dynamic model of proton exchange membrane fuel cells (PEMFCs). The multiple-input single-output (MISO) nonlinear model of the PEMFC is derived first. The dynamic model is then transformed into a multiple-input multiple-output (MIMO) square system by adding additional states and outputs so that the linearization and input-output decoupling approach can be directly applied. A PI tracking control is also introduced to the state feedback control law in order to reduce the steady-state errors due to parameter uncertainty. This paper also proposes an adaptive genetic algorithm (AGA) for the multi-objective optimization design of the tracking controller. The comprehensive results of simulation demonstrate that the PEMFC with nonlinear control has better transient and steady-state performance compared to conventional linear techniques.

Nowadays, the popular renewable energy sources include wind power, solar power generation and fuel cells. However, wind power and solar power are usually affected by external environmental factors, which cause the instability of the power generator output. In contrast to wind and solar power, fuel cells generate electricity stably and are less susceptible to external environment factors [

A PEMFC is a nonlinear and strongly coupled dynamic system. As the driven load changes, the output current changes and the electrochemical reaction is simultaneously accelerated. If the inlet flow rate of oxygen in the cathode is too low, the output power of PEMFC system would be decreased because of a lack of oxygen, which is known as starvation. In order to generate a reliable and efficient power response and prevent detrimental degradation of the stack voltage, it is very important to design an effective control strategy to achieve optimal oxygen and hydrogen inlet flow rates control.

Many control strategies been adopted nowadays for controlling PEMFC systems. Golbert [

State feedback linearization and input-output decoupling for nonlinear dynamic models have been widely used to enhance transient performance [

The working process of a PEMFC is accompanied with liquid/vapor/gas mixed flow transportation, heat conduction and electrochemical reactions. In order to simplify the analysis, several assumptions are made as listed below:

The governing equation is the Nernst equation.

The entire PEMFC is at the same operating temperature.

The entire gas is the ideal gas at a relative humidity of 100%.

The electrolyte membrane is of a high proton conductivity.

The gases are completely pure hydrogen and oxygen.

The nonlinear dynamic model developed in this paper is based on the FC models presented in [

The output voltage of a single fuel cell, according to the Nernst equation, is formulated as:

In the above equation _{Nernst} denotes the thermodynamic potential, that is the reversible voltage of the cell, represented by:
_{0} the universal gas constant (8.315 J/mol K), _{H2} and _{O2} are the partial pressures of hydrogen and oxygen respectively, T and T_{0} are the cell's operating temperature and reference temperature respectively.

_{act} represents the activation voltage drop, which is the polarization arising from the cathode and anode, given as:
_{j} the semi-empirical coefficients, defined as:

According to Henry's law, the concentrations of both the hydrogen and oxygen on the catalyst surfaces of anode and cathode are given as:

_{ohmic} represents the voltage drop across (_{M}, _{C}), that is, the equivalent resistance of the proton exchange membrane and an external circuit respectively, expressed as:

It is not an easy task to estimate in advance the value of _{C} over the range of PEMFC working temperatures, so it is treated as a constant in most cases. _{con} represents the concentration polarization voltage drop caused by mass transfer of reactant gas, which can be used to indicate the fuel cell voltage loss resulted from the high-current operating, written as:
^{2}) the cell current density, _{max} is the maximum current density ranging between 500 and 1500 mA/cm^{2}.

Substituting

Inasmuch as the reformer outputs the fuel rate, rather than the gas pressure required in the simulation model, there is a need to convert this flow rate into the gas pressure. As put forward in [_{a}_{c}

By using of the ideal gas law,

Firstly, consider the following MIMO affined nonlinear system:

Since the number of outputs is less than that of inputs in the above nonlinear model, the decoupling matrix in the feedback linearization is not a square matrix,

The addition of two extra states _{3} and _{4} and two extra outputs _{2} and _{3} converts the MISO nonlinear system,

The objective of state feedback exact linearization is to create a linear differential relation between the output

The approach in obtaining the exact linearization of the MIMO systems is to differentiate the output _{j}_{f}h_{j}_{j}

Similarly, in the case of another vector field _{i}

Assuming that _{j}

A substitution of

In case

For convenience, assuming that:

The inverse of

In the end, a substitution of

Substituting

This is a linear and input-output decoupling system. Comparing _{2} and _{3} are the same as _{2} and _{3}. Accordingly, _{1} is the only quantity which can be used for tracking control. In this form of the nonlinear control, a tracking error may exist due to parameter uncertainty. To obtain a more robust control, a PI controller is applied as in [_{1} = _{1} − _{1}_{ref}_{up}_{ui}_{1} is represented as:

As derived in the preceding section, an original nonlinear system is converted into a linear and input-output decoupling system. Besides, the control performance can be improved by the addition of a PI controller into a feedback control law _{1}. The control law _{1} mostly adjusts the inlet flow rate of hydrogen from the reformer, while the oxygen flow rate is dependent on the flow ratio τ_{H–O} between hydrogen and oxygen [

Proven more efficient than conventional algorithms, genetic algorithms were developed as a random search approach to locate the global optimum. However, in consideration of the distinct nature of search problems, a simple GA is not expected to find the global optimum as intended [

To demonstrate the performance of the proposed nonlinear control law, a Matlab/Simulink is used to build the PEMFC system dynamic model with nonlinear controller. In this work, the simulation parameters adopted are those of a single Ballard Mark V PEMFC. Hydrogen is employed as the fuel, oxygen is the oxidant, and a Nafion 117 PEM (Walther Grot of DuPont, Wilmington, DE, USA) is employed as well. All the cell parameters are tabulated in

To compare the efficiency of the proposed nonlinear controller, the conventional PID controller is also implemented for the PEMFC system. All the PID control parameters had been determined ahead of the simulation. Employing the Ziegler-Nichols rule to tune such parameters, as the first step, setting _{i}_{d}_{p}_{p}_{p}_{p}_{i}_{d}

The load current is changed for testing the transient behaviors of PEMFC with nonlinear control.

Even though a superior control performance is seen, the PI tracking controller parameters, determined by the Ziegler-Nichols rule, are not necessarily the optimal ones. For this sake, the following is devoted to the search of the optimal control parameters and the performance comparison. Tabulated in

Consequently, the optimal parameters obtained are _{up}_{ui}

Plotted in

A nonlinear control strategy utilizing the linearization and input-output decoupling approach is proposed in this paper for nonlinear control of PEMFCs. A MIMO dynamic nonlinear model of a PEMFC appropriate for developing the nonlinear controller is also presented. By adding a tracking controller to the state feedback control law, which is optimally designed by AGA, the steady-state errors due to parameter uncertainty can be effectively reduced. The comprehensive simulation results demonstrate that the PEMFC with nonlinear control has better transient and steady-state performance compared to conventional linear techniques. The proposed nonlinear control strategy and dynamic nonlinear model have the potential to become valuable tools for modeling and control of PEMFC systems.

The research was supported by the National Science Council of the Republic of China, under Grant No. NSC 101-ET-E-167-003-ET.

The authors declare no conflict of interest.

Block diagram of the proposed PEMFC nonlinear control with linearization and input-output decoupling.

Block diagram of the PEMFC with optimal tuning of PI controllers using AGA.

Variation of load current step changes.

Variation of output voltage.

Variation of output voltage error.

Variation of output power.

Variation of hydrogen flow rate.

Variation of oxygen flow rate.

Variation of hydrogen pressure.

Variation of oxygen pressure.

An output voltage comparison between before and after optimized feedback linearization control.

An enlarged view of

Parameters of the Ballard Mark V fuel cell.

343.15 K | _{1} |
−0.948 | |

50.6 cm^{2} |
_{2} |
(286 + 20 ln _{H2}) × 10^{−5} | |

λ | 178 μm | _{3} |
7.6 × 10^{−5} |

_{H2} |
1 atm | _{4} |
−1.93 × 10^{−4} |

_{O2} |
1 atm | Ψ | 23 |

0.016 V | _{max} |
150 mA/cm^{2} | |

_{C} |
0.0003 Ω | _{n} |
1.2 mA/cm^{2} |

Parameters adopted when performing an AGA.

Searching Range | _{p} |
0∼100 |

| ||

_{i} |
0∼100 | |

| ||

Population Size | 50 | |

Generation Number | 20 | |

Bit Number | 30 | |

Crossover Rate | 0.9 | |

Mutate Rate | 0.03 |