*3.3. RL Model Parameters*

Using the procedure shown in Section 2.4, which describes the method for determining the parameters of the equivalent electric model RL, and using the experimental responses shown in Figures 8 and 9, we obtained the results listed in Table 2. Figure 10 shows a comparison between the experimental voltage of the fuel cell and that obtained from the RL model. One can show a good fit for the proposed RL model.

**Table 2.** RL circuit parameters.


**Figure 10.** Comparison between the experimental and RL models. **Figure 10.** Comparison between the experimental and RL models.

### **4. Comparison between Different Models 4. Comparison between Different Models**

In this section, we will evaluate the static and the dynamic behaviour of the studied models, and compare them to the experimental data of the used Nexa 1200 fuel cell module. All models are simulated using MATLAB/Simulink software. As a load of these models, we used a controlled current source whose variations were programmed similarly to those used for the experiment. Figure 11 illustrates the simulated models, while the experiments were carried out according to Figure 7. All of the parameters used for the simulation models are listed in Table 3. In this section, we will evaluate the static and the dynamic behaviour of the studied models, and compare them to the experimental data of the used Nexa 1200 fuel cell module. All models are simulated using MATLAB/Simulink software. As a load of these models, we used a controlled current source whose variations were programmed similarly to those used for the experiment. Figure 11 illustrates the simulated models, while the experiments were carried out according to Figure 7. All of the parameters used for the simulation models are listed in Table 3.

**Table 3.** Simulation parameters.

**Table 3.** Simulation parameters.

Generic MATLAB mode (GMM)

RC model (RCM); and (**d**) RL model (RLM).


Voltage at 0 A and 1 A [30] Nominal operating point (52 A, 24.23 V) Maximum operating point (100 A, 20 V)

(**d**)

F 130 = ܥ

**Type of Model Parameters**  Nonlinear model (NLM) Parameters used in [1]

RC model (RCM) <sup>ܧ</sup> <sup>=</sup> 28.32 <sup>V</sup>; ܴ <sup>=</sup> 2.89958 ݉Ω; ܴ <sup>+</sup> ܴை <sup>=</sup> <sup>155</sup> ݉Ω;

Proposed RL model (RLM) ܧ = 28.32V ; ܴ<sup>ଵ</sup> = 157.70 ݉Ω; ܴ<sup>ଶ</sup> = 156.17 ݉Ω;ܮ = 3.1078 H

**Figure 11.** Simulated models in MATLAB/Simulink: (**a**) nonlinear model (NLM); (**b**) generic MATLAB model (GMM); (**c**)

(**c**)

**Figure 10.** Comparison between the experimental and RL models.

0 50 100 150 200 250 300 350 400

time (s)

Fuel cell voltage vfc

(V)

Experimental RL model

**4. Comparison between Different Models**

22

23 24

25 26

27 28 29

simulation models are listed in Table 3.

**Figure 11.** Simulated models in MATLAB/Simulink: (**a**) nonlinear model (NLM); (**b**) generic MATLAB model (GMM); (**c**) RC model (RCM); and (**d**) RL model (RLM). **Figure 11.** Simulated models in MATLAB/Simulink: (**a**) nonlinear model (NLM); (**b**) generic MATLAB model (GMM); (**c**) RC model (RCM); and (**d**) RL model (RLM). *Micromachines* **2021**, *12*, x 12 of 15

### **Table 3.** Simulation parameters. *4.1. Static Behaviour 4.1. Static Behaviour*

**Type of Model Parameters**  Nonlinear model (NLM) Parameters used in [1] The static behaviour of the simulated models is compared to the experimental results. Figure 12 illustrates the obtained current–voltage characteristics. The static behaviour of the simulated models is compared to the experimental results. Figure 12 illustrates the obtained current–voltage characteristics.

Voltage at 0 A and 1 A [30]

In this section, we will evaluate the static and the dynamic behaviour of the studied models, and compare them to the experimental data of the used Nexa 1200 fuel cell module. All models are simulated using MATLAB/Simulink software. As a load of these models, we used a controlled current source whose variations were programmed similarly to those used for the experiment. Figure 11 illustrates the simulated models, while the experiments were carried out according to Figure 7. All of the parameters used for the

**Figure 12.** *i–v* characteristics of simulated models compared to the experiments. **Figure 12.** *i–v* characteristics of simulated models compared to the experiments.

To compare between different models, the following root-mean-square error (RMSE) criterion is selected, which is a frequently used measure of the differences between values predicted by a model and the values observed: To compare between different models, the following root-mean-square error (RMSE) criterion is selected, which is a frequently used measure of the differences between values predicted by a model and the values observed:

$$RMSE = \sqrt{\frac{\sum\_{k=1}^{N} \left(V\_{f\text{cx}}(k) - V\_{f\text{cm}}(k)\right)^2}{N}} \tag{28}$$

where (ܸ௫ − ܸ) is the error between the measured (experimental) and the modelled fuel cell voltage; and *N* is the total number of samples. The obtained results are summarized in Table 4. It is evident from the table that the nonlinear model is a better model; however, it requires a longer computational time, which is considered a drawback for real-time application and control purposes. The classic RC model and the proposed RL model have practically the same RMSE in static conditions. Nevertheless, we will later where (*Vf cx* − *Vf cm*) is the error between the measured (experimental) and the modelled fuel cell voltage; and *N* is the total number of samples. The obtained results are summarized in Table 4. It is evident from the table that the nonlinear model is a better model; however, it requires a longer computational time, which is considered a drawback for real-time application and control purposes. The classic RC model and the proposed RL model have

> Nonlinear model (NLM) 0.1852 Generic MATLAB model (GMM) 0.1961 RC model (RCM) 0.2382 Proposed RL model (RLM) 0.2319

In this section, a comparison between the dynamic behaviour of four models and the experimental results in the presence of fuel cell current changes was studied. Figure 13 illustrates the dynamic behaviour of each model. As is clearly shown, the proposed RL equivalent electrical model presents the best dynamic behaviour compared to the conventional RC model used in the literature. The main advantages of this model lie in its

**Table 4.** RMSE criteria for static (*i–v*) characteristic.

*4.2. Dynamic Behaviour*

see the great supremacy of the proposed RL model in a dynamic regime.

simplicity and stability to produce the same behaviour as the fuel cell.

practically the same RMSE in static conditions. Nevertheless, we will later see the great supremacy of the proposed RL model in a dynamic regime.

**Table 4.** RMSE criteria for static (*i–v*) characteristic.

