## 2 Methodology
### Optimization Framework  
With strong robustness and low model dependence, evolutionary algorithms are applicable to optimization problems in nonconvex and discontinuous design spaces with high dimensional variables. The GA is a global optimization method that is constructed by imitating the genetic principles of nature species and it has been widely used in the field of optimization design for aircraft systems [42; 43; 44; 45; 46]. A binary string is used in order to simulate the gene sequence of an individual. The optimization evolution of the population genome during reproduction is simulated by processing the binary sequences of the sample group in the iterative process. The primary population is formed with randomly generated binary strings. The binary strings are converted into real values of the variables in the design space by linear interpolation. The response obtained from the analysis is used in order to evaluate the individual fitness value. Evolutionary processes include selection, replication, crossover, and variation. The optimal population is obtained through iterations. The GA optimization framework utilized in this study was established by referring to Matlab toolbox Gatbx [47; 48]. The optimization problem in this study is to search the optimal variables to maximize the objective function, which can be formulated as  
\[\left\{ \begin{array}{c}\max F(v)=\eta=\left(1-\frac{P_{Active}}{P_{Baseline}} \right)\times 100\%\\ \text{s.t.}\;v_{i}^{lower}\leq v_{i}\leq v_{i}^{upper},\;i=1,2,3,\ldots,n_{dv} \end{array} \right. \tag{5}\]  
where \(F(v)\) is the objective function (the power reduction), \(v\) is the variable vector, \(\eta\) is the rotor power reduction, \(P_{Active}\) is the rotor power with active twist control, \(P_{Baseline}\) is the power of the baseline rotor, \(v_{i}^{lower}\) and \(v_{i}^{upper}\) are the lower and upper bounds of the variables, respectively, and \(n_{dv}\) is the dimension of the variables. The GA optimization is performed in order to determine the optimal active twist control schedules and the corresponding segment layouts. The optimization variables are the amplitudes of sine and cosine components of active twist rate and spanwise positions of the segment points. The rotor power in a steady-level flight is predicted, and the power reduction is converted for evaluating the individuals. Figure 8 describes the flowchart of the GA optimization. A parametric study of active twist control in Ref. [25] shows that the rotor \(\mathrm{L/D_{e}}\) varies linearly with the amplitude of twist control, but the structural loads vary nonlinearly. At high amplitudes, the twist control structural loads significantly increase. Meanwhile, high amplitudes call for high drive power. The twist deformation with existing actuator techniques for active twist rotor is relatively limited. Therefore, an appropriate range of active twist rate should be determined in combination with engineering practice. The maximum achievable active twist rate is approximately 3.0\({}^{\circ}\)/m peak-to-peak, as shown in experimental and numerical studies [49]. A parametric study of the amplitudes for uniform twist control is performed in the present study, as shown in following section. There is a linear increase in power reduction with the increase in amplitudes. However, the increase begins to plateau with the amplitude beyond 1.0\({}^{\circ}\)/m (2.0\({}^{\circ}\)/m peak to peak). Therefore, a saturation limit of the amplitudes of active twist rate is set as 1.0\({}^{\circ}\)/m (2.0\({}^{\circ}\)/m peak to peak) in this study. The first type of variables defines the amplitude of the active twist control. The range of coefficients for Fourier series is set from \(-1.5^{\circ}\)/m to 1.5\({}^{\circ}\)/m with a resolution of 0.1\({}^{\circ}\)/m in order to obtain a wider fitting capability. Each coefficient is mapped with a 5-bit binary number in the GA. The second type of variables is used to represent the segment layouts. The positions of the segment points determine the segment layout.