**3. Results**

Figure 8 tackles the response behavior of the stack current signal under the application of the proposed MPC method and the classical PI control. To test the performance of the controllers and their capability of counteracting the disturbance, load resistance variation is applied at two times instances *t*<sup>1</sup> = 25 s and *t*<sup>2</sup> = 45 s. These times correspond, respectively, to resistance rising from 20 to 50 Ω and decreasing from 50 to 20 Ω. The coeffi-

cient parameters of the PI controller were tuned through the minimization of *IAE*, and they are equal to 0.02 and 10 for the proportional and integral terms, respectively.

**Figure 8.** (**a**) Stack current signal; (**b**) stack current behavior when increasing the load resistance; (**c**) stack current behavior when decreasing the load resistance; (**d**) steady state.

It is clear from the first load variation, depicted in Figure 8b, that the MPC approach converges rapidly to the reference current with a response time equal to 1.3 s against an important response value for the classical PI controller, which is around 6.8 s. It should be noted that 0.12 s of the response time was caused by the delay time, which occurred at the moment in which the load variation was applied. Hence, the proposed MPC controller achieved a significant improvement in the convergence speed of almost 81%. On the other hand, the MPC presents a reduced undershoot equal to 1.73 A compared with the conventional PI method, which is around 2.1 A. Consequently, the proposed algorithm can effectively reduce the undershoot with an enhancement of 17.61% compared with the PI controller.

The impact of reducing the load resistance on the response of the stack current is illustrated through Figure 8c. It is obviously clear from this figure that the PI controller takes a significant time to reach the current reference with a response time equal to 7.25 s, while only 0.51 s is obtained via the proposed MPC, which effectively outperforms the convergence speed of the PI with 92.9%. According to this figure, it is noticed that the current signal controlled via the proposed MPC made a delay time of 0.02 s. However, this time is almost negligible, and it has no negative effect on the response time. Regarding the overshoots, a significant one of almost 3.65 A is shown on the response behavior of the conventional PI, while an improvement of around 13.69% on the overshoot is obtained using the proposed MPC method.

Figure 9a–c illustrates, respectively, the real-time response of the PEMFC voltage, power and duty cycle delivered by the classical PI and the proposed MPC approach. The slight variation between the experimental test of the PI and MPC that appeared in a,b

and c occurred due to the effect of the operating temperature on the membrane since it is difficult to carry out two experiments at exactly the same temperature. It should be noted that this variation did not appear in the graphs of the stack current (Figure 8) since it is a controlled signal where both of the algorithms drive the stack current to operate at the same reference current *Iref* .

**Figure 9.** (**a**) PEMFC stack voltage signal; (**b**) PEMFC stack power; (**c**) duty cycle signal.

According to Figure 9a, the effectiveness of the proposed MPC algorithm over the conventional PI appears to reduce the overshoots and undershoots of the stack voltage. Thus, the PI controller presents a voltage value around 1.11 V and 1.33 V for the first and the second load variation, respectively. On the other hand, the proposed MPC shows values of 0.98 V and 1.46 V for the same load variations.

From Figure 9b, it can be seen that the proposed MPC method effectively tracks the desired output power of the PEMFC with an almost negligible ripple around the steady state. Moreover, in comparison with the conventional PI controller, the results show that a reduction of 4.18 W and 3.73 W in the undershoot and overshoot are obtained for the first and the second load variation, respectively.

The real-time responses of the output current, voltage and power for the DC–DC boost converter are depicted in Figure 10a–c. The latter clearly shows the impact of the variable load resistance on the response behavior of the output current and the output voltage for the two controllers. Furthermore, the slow converging and high overshoots of the PI controller in comparison with the proposed MPC are clearly presented in this figure.

**Figure 10.** (**a**) DC–DC output current signal; (**b**) DC–DC output voltage signal; (**c**) DC–DC output power signal.

Finally, it is clearly demonstrated in the above results that the proposed MPC has succeeded in overcoming the drawbacks of the conventional PI controller. Hence, a robust and fast response, as well as better dynamic behavior when facing large load variation, are obtained via the application of the proposed MPC method.
