2.4.2. Determination of Model Parameters

2.4.2. Determination of Model Parameters The electrical model presented in Figure 6 involves three parameters— ܮ, ܴ<sup>ଵ</sup> , and ܴଶ—which can be determined using the response of the fuel cell voltage to load current step change. Let us first introduce some useful electrical relationships based on the pro-The electrical model presented in Figure 6 involves three parameters*L*, *R*1, and *R*2—which can be determined using the response of the fuel cell voltage to load current step change. Let us first introduce some useful electrical relationships based on the proposed circuit.

posed circuit. The fuel cell voltage is governed by the following equations:

The fuel cell voltage is governed by the following equations:

$$V\_{fc} = E\_{oc} - R\_2 I\_{fc} - R\_1 \left( I\_{fc} - I\_L \right) \tag{19}$$

$$L\frac{dI\_L}{dt} = R\_1 \left(I\_{fc} - I\_L\right) \tag{20}$$

$$L\frac{dI\_L}{dt} + R\_1 I\_L = R\_1 I\_{fc} \tag{21}$$

For *t* < *t*<sup>0</sup> we suppose that *If c* = *I*<sup>0</sup> = *cte*, which corresponds to a constant fuel cell voltage *Vf c* = *V*<sup>0</sup> = *cte* (see Figure 5). At the instant *t* = *t*0, we apply a current step change from *I*<sup>0</sup> to *I*1, and then the inductor current *I<sup>L</sup>* will evolve, using Equation (21), according to the following equation:

$$I\_L = I\_1 + (I\_0 - I\_1)e^{-\frac{(t-t\_0)}{\tau\_L}} \tag{22}$$

where *τ<sup>L</sup>* = *<sup>L</sup> R*1 is a time constant of the RL circuit. Moreover, using Equations (19) and (22), the voltages *V*01, *V*02, and *V*<sup>∞</sup> represented in Figure 5 are given as follows:

$$V\_{01} = E\_{oc} - R\_2 I\_0$$

$$V\_{02} = E\_{oc} - R\_2 I\_1 - R\_1 (I\_1 - I\_0) \tag{23}$$

$$V\_{\infty} = E\_{oc} - R\_2 I\_1$$

It follows that the voltage variations ∆*V*<sup>1</sup> and ∆*V*2, corresponding to the current variation ∆*I* = *I*<sup>1</sup> − *I*0, are given by:

$$
\Delta V\_1 = V\_{01} - V\_{02} = (R\_1 + R\_2)(I\_1 - I\_0) = (R\_1 + R\_2)\Delta I \tag{24}
$$

$$
\Delta V\_2 = V\_{01} - V\_{\infty} = R\_2(I\_1 - I\_0) = R\_2 \Delta I \tag{25}
$$

Now, taking any instant *t*<sup>1</sup> > *t*<sup>0</sup> in the transient of the fuel cell voltage, using Equations (19), (22), and (23), one has:

$$V\_{t1} = V\_{\infty} - R\_1(I\_1 - I\_0)e^{-\frac{(t\_1 - t\_0)}{\tau\_L}} \tag{26}$$

∆*V*3

which, in turn, taking into account Equations (24) and (25), gives:

$$
\Delta V\_3 = V\_{\infty} - V\_{t1} = R\_1 \Delta I e^{-\frac{\Delta t}{\tau\_L}} = (\Delta V\_1 - \Delta V\_2)e^{-\frac{\Delta t}{\tau\_L}} \tag{27}
$$

where ∆*t* = *t*<sup>1</sup> − *t*0.

Finally, the procedure for determining the RL model parameters can be summarized as follows:

