6.2.2. Bat/SC Sizing

The sizing process of this source configuration is similar to the previous one. The supercapacitor feeds the motor in high-power segments, especially during take-off and landing. During the cruising segment, there is the possibility of charging the supercapacitor with the excess energy supplied by the battery. The power of the supercapacitor and the battery is related to the power of the motor by

$$\begin{cases} P\_{\rm SC} = \mathbf{x} \cdot P\_{\rm mot} \\ P\_{\rm bat} = (1 - \mathbf{x}) \cdot P\_{\rm mot} \end{cases} \tag{48}$$

where x is the coefficient of the supercapacitor power to the motor power. The global flight time of the aerial vehicular *tflight* is obtained by the contribution of the supercapacitor *tflightSC* and the battery *tflightBat*. It is given by

$$t\_{flight} = t\_{flightSC} + t\_{flightBat} \cdot \tag{49}$$

From Equations (15) and (23), the flight time is given by

$$t\_{flight} = \left(\frac{m\_{SC} \cdot \rho\_{SC} \cdot \eta\_c \cdot \eta\_S \text{C}}{\text{x}} + \frac{m\_{bat} \cdot \rho\_b \cdot \eta\_c \cdot \eta\_b}{1 - \text{x}}\right) \cdot \left(\frac{\eta\_{MP}}{(m\_{SC} + m\_{bat} + m\_{other}) \cdot \text{g}}\right) \cdot \text{(50)}$$

The flight time evolution in terms of the battery mass and the supercapacitor mass is given by Figure 16. This evolution is obtained for a hybridization coefficient of *x* = 5%.

**Figure 16.** Flight time evolution in terms of the supercapacitor mass *mSC* and the battery mass *mbat*.

In the case of the Bat/SC configuration, maximizing flight time is more influenced by increasing the battery mass rather than increasing the supercapacitor mass. This is because the supercapacitor has a lower specific energy compared to the battery. As a result, when the battery mass increases, the overall autonomy of the energy storage system improves. It is also remarkable that the flight time evolution as a function of the battery mass and the supercapacitor mass, for a fixed hybridization coefficient *x*, presents a single basin of attraction. The maximum in this case should be reached in the permitted region imposed by the *GTOW*. The energy storage optimal masses and the hybridization coefficient, allowing the maximization of the multirotor aerial vehicle flight time, are located using a global non-linear optimization. The optimization problem in this case is given by

$$\begin{cases} \max(\mathbf{t}\_{flight}) = \min(-\mathbf{t}\_{flight}),\\ 0 < m\_{bat} + m\_{SC} \le GTOW - m\_{other}. \end{cases} \tag{51}$$

Figure 17 gives an example of this optimization for a hybridization coefficient of *x* = 5%. Through this optimization, a flight time of *tflight* = 14.27 min is obtained. The sizing of the battery remains similar to the previous case. The optimized supercapacitor mass allows obtaining the required energy based on the Maxwell cell energy density reported in Table A2 in the Appendix A. Thus, the SC capacity must satisfy the following condition:

$$C\_{SC} \geq \frac{16}{3} \frac{E\_{SC}}{U\_{SC}} \,\mathrm{}\tag{52}$$

The optimized parameters of the battery and the SC are reported in Table A6.

**Figure 17.** Example of a flight time optimization in Bat/SC configuration case.
