**4. Motor/Propeller Optimization Problem**

The optimization technique is based on the simulated annealing algorithm *SAA*, which was introduced by inspiring the annealing procedure of the metalworking. In a general manner, the SA algorithm adopts an iterative movement according to the variable temperature parameter, which imitates the annealing transaction of the metals [37]. This algorithm is directly explored using the MATLAB global optimization toolbox. The efficiency of the pair motor/propeller *ηMP* is used as an objective function in this case. It is given by

$$
\eta\_{MP} = \frac{T\_{flight}}{P\_{mflight}} \,\tag{26}
$$

where *Tflight* and *Pmflight* are, respectively, the propeller thrust and the motor power during the flight operation. The motor power is given by

$$P\_{mflight} = \mathcal{U}\_{mfllight} \cdot I\_{mfllight} \tag{27}$$

from the propeller model presented in Equation (1), the propeller velocity and the propeller torque during the flight are given by

$$\begin{cases} N\_{flight} = \frac{60}{D\_p^{\prime 2}} \cdot \sqrt{\frac{T\_{flight}}{\rho \cdot \mathcal{C}\_T}}\\ M\_{flight} = \frac{\mathcal{C}\_M \cdot D\_p}{\mathcal{C}\_T} \cdot T\_{flight} \end{cases} \tag{28}$$

and from the motor model presented in Equation (9), the motor current and the motor voltage are given by

$$\begin{cases} I\_{m0} \approx 0, \\ I\_{mflight} = \frac{\pi \cdot \mathbb{C}\_{M} \cdot D\_{p}}{\Re \mathrm{C}\_{T} \cdot K\_{E}} \cdot T\_{flight}, \\ \mathrm{LI}\_{mflight} = \frac{\pi \cdot \mathbb{C}\_{M} \cdot D\_{p}}{\Re \mathrm{C}\_{T} \cdot K\_{E}} \cdot \mathrm{R}\_{m} \cdot T\_{flight} + \frac{60 \cdot \mathrm{K}\_{P}}{\mathrm{D}\_{p}^{2}} \cdot \sqrt{\frac{T\_{flight}}{t^{\gamma} \cdot \mathrm{C}}}. \end{cases} \tag{29}$$

Thus, the optimization objective function expression is given by

$$\eta\_{MP} = \frac{1}{\left(\frac{\pi \cdot \mathbb{C}\_{M} \cdot D\_{p}}{\Re \mathbb{O} \cdot \mathbb{C}\_{T} \cdot \mathbb{K}\_{E}}\right)^{2} \cdot T\_{flight} \cdot R\_{m} + \frac{2\pi \cdot \mathbb{C}\_{M}}{D\_{p}} \cdot \sqrt{\frac{T\_{flight}}{\rho \cdot \mathbb{C}\_{T}}}} \cdot \tag{30}$$

For a fixed thrust imposed by the *GTOW*, the motor/propeller efficiency evolution in terms of propeller parameters is given in Figure 8. Through this figure, it is noticeable that the motor/propeller efficiency presents a single attraction basin, enabling the rapid identification of the point that maximizes the objective function.

**Figure 8.** Motor/propeller efficiency evolution in terms of propeller parameters,

In order to avoid the motor overheating during the flight, which could influence the propulsion chain efficiency, the motor current and voltage must remain below their maximum values, *UmMax* and *ImMax*, imposed by the motor design:

$$
\mathcal{U}I\_m \le \mathcal{U}I\_{mMax} \text{ \& } I\_m \le I\_{mMax} \text{\tag{31}}
$$

which leads to establishing the following constraint on the propeller velocity and torque:

$$\begin{cases} \mathcal{N}\_{\text{max}} = \frac{lI\_{m\text{Max}} - R\_{m\text{Max}} \cdot I\_{m\text{Max}}}{K\_E},\\ \mathcal{M}\_{\text{max}} = \frac{30 \cdot (I\_{m\text{Max}} - I\_{m0}) \cdot K\_E}{\pi}, \end{cases} \tag{32}$$

thus, the propeller diameter must remain below its maximum value *DPmax* imposed by the motor overheating avoidance condition:

$$D\_p \le D\_{P\max} = \left(M\_{\max} \, ^4 \cdot \left(\frac{60}{N\_{\max}}\right)^2 \cdot \frac{1}{C\_M \cdot \rho}\right)^{\frac{1}{5}}.\tag{33}$$

Thus, the optimization problem of the motor/propeller is given as follows:

$$\begin{cases} \max(\eta\_{MP}) = \min(-\eta\_{MP}),\\ 0 < D\_p \le D\_{p\text{MaxElec}\prime} \\ 0 < \varphi\_p \le \pi. \end{cases} \tag{34}$$
