6.2.1. Bat/HFC Sizing

Depending on the flight mission segment, the motor power (load) is supplied by the hydrogen fuel cell or the battery. In both cases, the battery and the hydrogen fuel cell power are related to the motor power by

$$\begin{cases} P\_{bat} = \mathbf{x} \cdot P\_{mat\prime} \\ P\_{HFC} = (1 - \mathbf{x}) \cdot P\_{mat\prime} \end{cases} \tag{43}$$

where *x* is the hybridization coefficient of the battery power to the motor power. The global flight time of the aerial vehicle *tflight* is obtained by the contribution of the battery *tflightBat* and the hydrogen fuel cell *tflightHFC*. It is given by

$$t\_{flight} = t\_{flight} + t\_{flightHFC} \cdot \tag{44}$$

From Equations (15) and (21), the flight time is given by

$$t\_{flight} = \left(\frac{m\_{bat} \cdot \rho\_b \cdot \eta\_c \cdot \eta\_b}{\chi} + \frac{m\_{H\_2} \cdot LHV \cdot \eta\_c \cdot \eta\_{FC}}{1 - \chi}\right).$$

$$\left(\frac{\eta\_{MP}}{\left(m\_{STAT} + m\_{bat} + m\_{others}\right) \cdot \mathcal{g}}\right). \tag{45}$$

From this equation, it is evident that the flight of the multirotor aerial vehicle is influenced by the mass of hydrogen *mH*<sup>2</sup> , the mass of the battery *mbat*, and the hybridization coefficient *x*. The choice of this coefficient depends on the duration of the flight mission segments during which the maximum power is required. These segments typically represent less than 14% [41,42] of the total flight duration. The flight time variation with respect to the hydrogen mass, HFC stack mass, and battery mass is illustrated in Figure 14.

**Figure 14.** Flight time evolution in terms of the HFC stack mass *mSTACK* and the battery mass *mbat*.

It is remarkable that maximizing the flight time is more favored by increasing the hydrogen mass than increasing the battery mass. This is due to the fact that as the hydrogen mass increases, the specific energy of the system also increases, resulting in an extended flight time.

It is also noticeable that the flight time evolution—as a function of hydrogen mass and the battery mass for the fixed hybridization coefficient *x*—presents a single basin of attraction. The maximum in this case is not attainable because the mass of the energy sources is limited by the constraint of the *GTOW*.

In order to locate the energy sources' optimal masses, which allow for maximizing the flight time with the constraint of the *GTOW* for different values of the hybridization coefficient *x*, a nonlinear global optimization was carried out. The algorithm considered in this part is the same one that was used in the motor/propeller pair optimization part. The optimization problem in this case is given in

$$\begin{cases} \max(t\_{flight}) = \min(-t\_{flight}),\\ 0 < m\_{ball} + m\_{STAK} \le \text{GTOW} - m\_{other}. \end{cases} \tag{46}$$

Figure 15 gives an example of this optimization for a hybridization coefficient of *x* = 14%. In this configuration, the obtained battery mass makes it possible to compute the battery capacity using the equation reported in (16). Based on the required capacity, the number of battery-parallel branches is computed. Regarding the sizing of the HFC in this case, there is the tank sizing, which is defined by hydrogen mass obtained by the optimization part, using the regression model presented in Equation (23). The stack sizing, or the fuel cell area sizing, depends on the hybridization coefficient x, by using the following equation:

$$\begin{cases} P\_{\rm FC} = (1 - \chi) \cdot N\_{\rm m} \cdot \frac{P\_{\rm m}}{\eta\_{\rm c} \cdot \eta\_{\rm FC}}. \\ A\_{\rm cellI} = \frac{P\_{\rm FC}}{\rho\_{\rm cellI} \cdot N\_{\rm cellI}} \end{cases} \tag{47}$$

Thus, the stack mass is deduced using Equation (22).

**Figure 15.** Example of a flight time optimization in the Bat/HFC configuration case.

Through this optimization, a flight time of *tflight* = 56.18 min is obtained. The optimized parameters of the battery and the HFC are reported in Table A5.
