**2. PEM fuel cell**

A PEM fuel cell can be described as a static device that converts the chemical energy of a fuel directly, isothermally, and continuously into electrical energy. In this process, only the reac‐ tion between hydrogen and oxygen occur. The only by-products are water and heat. Similarly toabattery,afuel cell consistsoftwoelectrodes (anodeandcathode)andanelectrolyte.Whereas astoragebatterycontainsallthesubstances intheelectrochemicaloxidation-reductionreactions involvedandhasthereforealimitedcapacity,thefuelcellissuppliedwithitsreactantsexternally and operates continuously as long as it is supplied with fuel. The basic scheme for a single cell is represented in Figure 1 and the reactions involved in the anode side, the cathode side and the overall reaction of the process are described by the equations 1 to 3 follows.

$$\rm H\_2 \to 2H^+ \text{+} 2e^- \tag{1}$$

The model selection differs for each application and user and the initial decisions are important to avoid changes later in the model evaluation process. The theoretical models are normally detailed, complex and usually require large computation time [8,9,11-12]. The semi-empirical models give a general voltage-current relationship without examining in depth the physical and electrochemical phenomena involved in the operation [1,5,6,10]. These models are usually

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach

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333

The electrical equivalent circuit represented Figure 2 corresponds to the semi-empirical model adopted for this study. This circuit is the electrical equivalent of the static and dynamic behaviour of the PEM fuel cell and includes the effects of the thermodynamic potential of the fuel cell and the losses. The equations 4 to 9 represent the static behaviour of the PEM while the dynamics is represented by equations 10 and 11. The capacitor C corresponds to the fuel cell phenomenon known as "charge double layer" on which the interface electrode/electrolyte acts as storage of energy element. The electrical power and efficiency are represented by

*V E VV V FC Nernst act Ohmic con* = -- - (4)

characterized by simple implementation and faster simulation.

equations 12 and 13 respectively.

Output voltage of one cell:

**Figure 1.** Scheme of a single cell.

$$2\text{H}^+ + \frac{1}{2}\text{O}\_2 + 2\text{e} \rightarrow \text{H}\_2\text{O} \tag{2}$$

$$\text{H}\_2\text{+}\bigvee\_2\text{O}\_2 \rightarrow \text{H}\_2\text{O}\tag{3}$$

#### **2.1. Modelling of the PEM fuel cell**

Many proton exchange membrane (PEM) fuel cell models have been investigated and presented in the literature [1-12]. The process of selecting the fuel cell model needs to clarify what are the necessary features to take into account in the model [7].

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach http://dx.doi.org/10.5772/54674 333

**Figure 1.** Scheme of a single cell.

The converter follows a resonant approach that provides low component stresses, high frequency operation, soft-switching commutation, and operation under a wide range of input

The control of the converter is divided into two parts, namely: i) the voltage controller, which is responsible for keeping constant the output voltage of the converter under loading variations and ii) the PEM controller, which is responsible for improving the performance by keeping the

The results are firstly presented for the PEM fuel cell and then for the whole system with load

The results demonstrate that the proposed converter is a good selection to improve the efficiency of PEM fuel cells because it allows an adequate control of the power delivered by the fuel cell while maintaining the requirements imposed by the load and minimizing the losses

A PEM fuel cell can be described as a static device that converts the chemical energy of a fuel directly, isothermally, and continuously into electrical energy. In this process, only the reac‐ tion between hydrogen and oxygen occur. The only by-products are water and heat. Similarly toabattery,afuel cell consistsoftwoelectrodes (anodeandcathode)andanelectrolyte.Whereas astoragebatterycontainsallthesubstances intheelectrochemicaloxidation-reductionreactions involvedandhasthereforealimitedcapacity,thefuelcellissuppliedwithitsreactantsexternally and operates continuously as long as it is supplied with fuel. The basic scheme for a single cell is represented in Figure 1 and the reactions involved in the anode side, the cathode side and the

2 2

Many proton exchange membrane (PEM) fuel cell models have been investigated and presented in the literature [1-12]. The process of selecting the fuel cell model needs to clarify

2 22 H+ O HO <sup>1</sup>

+ - H 2H +2e <sup>2</sup> ® (1)

<sup>2</sup> ® (3)

<sup>1</sup> 2H + O +2e H O <sup>2</sup> ® (2)

overall reaction of the process are described by the equations 1 to 3 follows.

+

what are the necessary features to take into account in the model [7].

and output conditions.

332 New Developments in Renewable Energy

disturbance.

PEM fuel cell in its optimal operating point.

by using soft-switching control.

**2.1. Modelling of the PEM fuel cell**

**2. PEM fuel cell**

The model selection differs for each application and user and the initial decisions are important to avoid changes later in the model evaluation process. The theoretical models are normally detailed, complex and usually require large computation time [8,9,11-12]. The semi-empirical models give a general voltage-current relationship without examining in depth the physical and electrochemical phenomena involved in the operation [1,5,6,10]. These models are usually characterized by simple implementation and faster simulation.

The electrical equivalent circuit represented Figure 2 corresponds to the semi-empirical model adopted for this study. This circuit is the electrical equivalent of the static and dynamic behaviour of the PEM fuel cell and includes the effects of the thermodynamic potential of the fuel cell and the losses. The equations 4 to 9 represent the static behaviour of the PEM while the dynamics is represented by equations 10 and 11. The capacitor C corresponds to the fuel cell phenomenon known as "charge double layer" on which the interface electrode/electrolyte acts as storage of energy element. The electrical power and efficiency are represented by equations 12 and 13 respectively.

Output voltage of one cell:

$$V\_{FC} = E\_{Normst} - V\_{act} - V\_{Olimic} - V\_{com} \tag{4}$$

**Figure 2.** Electrical equivalent circuit of the PEM fuel cell.

Thermodynamic potential:

$$E\_{\rm Nernst} = 1.229 - 0.85 \times 10^{-3} \times \left(T - 298.15\right) + + 4.31 \times 10^{-5} \times T \times \left[\ln\left(P\_{H2}\right) + \frac{1}{2}\ln\left(P\_{O2}\right)\right] \tag{5}$$

Activation over-potential:

$$V\_{act} = -\left[\xi \mathbf{1} + \xi \mathbf{2} \times T + \xi \mathbf{3} \times T \times \ln\left(\text{CO}\_2\right) + \xi \mathbf{4} \times T \times \ln\left(i\_{\text{FC}}\right)\right] \tag{6}$$

Ohmic over-potential:

$$V\_{\rm ohmic} = i\_{\rm FC} \left( R\_M + R\_{\rm C} \right) \tag{7}$$

Voltage across capacitor:

Electrical Time-constant:

Electrical power:

steady-state operation.

Efficiency:

t

1 1 *<sup>d</sup>*

*dt C*

*dV i V*

*act con*

*V V C Ra C R R C*

=´ =´ + =´ ç ÷

*FC d*

t

Methodology of Designing Power Converters for Fuel Cell Based Systems: A Resonant Approach

( ) *act con*

100%

1,48 *FC*

The circuit of Figure 3 corresponds to the experimental setup performed to obtain the electrical PEM characteristics. For each step of load the data is logged when the fuel cell achieved the

*f V*

h m

**2.2. Experimental tests made with the PEM Mark 1020**

**Figure 3.** Electrical circuit to test the PEM fuel cell Mark 1020.

æ öæ ö = ´ -´ ç ÷ç ÷ è øè ø (10)

*FC FC FC PiV* = ´ (12)

=´ ´ (13)

(11)

335

http://dx.doi.org/10.5772/54674

*FC*

è ø

æ ö +

*i*

Concentration over-potential:

$$V\_{con} = -B \times \ln\left(1 - \frac{I}{J\_{\text{max}}}\right) \tag{8}$$

Output voltage of the stack:

$$V\_s = \mathbf{n} \times V\_{\text{FC}} \tag{9}$$

Voltage across capacitor:

$$\frac{dV\_d}{dt} = \left(\frac{1}{C} \times i\_{\text{FC}}\right) - \left(\frac{1}{\tau} \times V\_d\right) \tag{10}$$

Electrical Time-constant:

$$
\sigma = \mathbf{C} \times \mathbf{R} \\
\mathbf{a} = \mathbf{C} \times \left( \mathbf{R}\_{\rm act} + \mathbf{R}\_{\rm con} \right) = \mathbf{C} \times \left( \frac{V\_{\rm act} + V\_{\rm com}}{i\_{\rm FC}} \right) \tag{11}
$$

Electrical power:

$$P\_{\rm FC} = \dot{\mathbf{i}}\_{\rm FC} \times V\_{\rm FC} \tag{12}$$

Efficiency:

Thermodynamic potential:

334 New Developments in Renewable Energy

**Figure 2.** Electrical equivalent circuit of the PEM fuel cell.

Activation over-potential:

Ohmic over-potential:

Concentration over-potential:

Output voltage of the stack:

( ) ( ) ( ) 3 5

x<sup>ù</sup> <sup>ë</sup> <sup>û</sup> (6)

( ) *V iRR ohmic FC M C* = + (7)

*V nV s FC* = ´ (9)

<sup>1</sup> 1.229 0.85 10 298.15 4.31 10 ln ln <sup>2</sup> *Nernst H O <sup>E</sup> <sup>T</sup> TP P* - - é ù = - ´ ´ - ++ ´ ´ ´ + ê ú

( ) ( ) <sup>2</sup> 1 2 3 ln 4 ln *Vact FC* =- + ´ + ´ ´ + ´ ´ é

ln 1 *con <sup>J</sup> V B*

*T T CO T i*

max

*J*

è ø

æ ö =- ´ - ç ÷

 x

xx

2 2

ë û (5)

(8)

$$
\eta = \mu\_f \times \frac{V\_{\text{FC}}}{1/48} \times 100\% \tag{13}
$$

#### **2.2. Experimental tests made with the PEM Mark 1020**

The circuit of Figure 3 corresponds to the experimental setup performed to obtain the electrical PEM characteristics. For each step of load the data is logged when the fuel cell achieved the steady-state operation.

**Figure 3.** Electrical circuit to test the PEM fuel cell Mark 1020.

#### *2.2.1. Output voltage and power*

The output voltage is measured by directly connecting the digital multimetter in parallel with the fuel cell. This is an uncontrolled DC voltage, which fluctuates with the load as well as with changes in the fuel input to the system. Figure 4 shows the stack voltage while Figure 5 corre‐ sponds to the stack power of the PEM Mark 1020. The results obtained for the both electrical variables are in accordance with the information provided by the manufacturer of the stack.

*2.2.2. Efficiency and hydrogen consumption*

**Figure 6.** Figure 6. Efficiency of the PEM Mark 1020.

**Figure 7.** Hydrogen consumed by the PEM Mark 1020.

The efficiency of PEM Mark 1020 is in the range of 40 % - 55 %, which minimum and maximum values are 45.15 % and 55.49 % respectively. The efficiency decreases slightly with the increase of the current density as is shown in Figure 6. The hydrogen consumed by the stack Mark 1020 is represented in Figure 7 below and is proportional to the power delivered by the fuel cell.

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**Figure 4.** Fuel cell voltage of PEM Mark 1020.

**Figure 5.** Electrical power of PEM Mark 1020.

#### *2.2.2. Efficiency and hydrogen consumption*

*2.2.1. Output voltage and power*

336 New Developments in Renewable Energy

**Figure 4.** Fuel cell voltage of PEM Mark 1020.

**Figure 5.** Electrical power of PEM Mark 1020.

The output voltage is measured by directly connecting the digital multimetter in parallel with the fuel cell. This is an uncontrolled DC voltage, which fluctuates with the load as well as with changes in the fuel input to the system. Figure 4 shows the stack voltage while Figure 5 corre‐ sponds to the stack power of the PEM Mark 1020. The results obtained for the both electrical variables are in accordance with the information provided by the manufacturer of the stack.

The efficiency of PEM Mark 1020 is in the range of 40 % - 55 %, which minimum and maximum values are 45.15 % and 55.49 % respectively. The efficiency decreases slightly with the increase of the current density as is shown in Figure 6. The hydrogen consumed by the stack Mark 1020 is represented in Figure 7 below and is proportional to the power delivered by the fuel cell.

**Figure 6.** Figure 6. Efficiency of the PEM Mark 1020.

**Figure 7.** Hydrogen consumed by the PEM Mark 1020.
