# Optimal Shape Design of Direct-Drive Permanent Magnet Generator for 1 kW-Class Wind Turbines
## 4 Optimal Shape Design  
### Design Space Exploration  
A design space exploration was conducted in order to identify design solutions that satisfy desired design requirements from a space of tentative design points resulting from applying optimizations at various levels of abstraction. To determine the optimal design parameters for the half model of the AFPM generator, as shown in Figure 6, an optimal shape design was carried out with weight W and active power P as the objective function and constraint, respectively. \[ \begin{split}&\text{Minimize W}=\text{W}\Big{(}\text{D}_{i}, \text{l},\text{b}_{1},\text{b}_{2},\text{h}_{1},\text{h}_{2},\text{N}_{\text{ ph}}\Big{)}\\ &\text{Subject to P}\Big{(}\text{D}_{i},\text{l},\text{b}_{1},\text{ b}_{2},\text{h}_{1},\text{h}_{2},\text{N}_{\text{ph}}\Big{)}\geq 250[\text{W}]\\ &\text{Design Parameters D}_{i},\text{l},\text{b}_{1},\text{b}_{2},\text{h}_{1},\text{h}_{2},\text{N}_{\text{ph}} \end{split}  \tag{11}\]  
The minimization of weight W and the requirement of active power P greater than or equal to 250 W were considered in this formulation. The optimal shape design consisted of seven design parameters, including six shape parameters such as \(\text{D}_{i},\text{l},\text{b}_{1},\text{b}_{2},\text{h}_{1}\) and \(\text{h}_{2}\), as shown in Figure 4, and the number of winding coil turns, \(\text{N}_{\text{ph}}\). To select specific parameter values, three levels of abstraction were adopted and specified in Table 3.  
An orthogonal array was utilized to estimate the effect of different design parameters on performance characteristics in a condensed set of analyses [28]. For this purpose, an appropriate orthogonal array such as L27(3\({}^{7}\)) was selected based on the number of design parameters and levels, indicating that only 27 experiments were required to study the effect of a maximum of seven variables at three levels each, instead of conducting 3\({}^{7}\) experiments. The 27 experiments for the design space exploration are listed in Table 4. For each experiment, the electromagnetic analysis under the rated-load condition was carried out, in which the active powers were calculated by induced voltages and currents using Equation (7). The resistive load values were considered to maximize active power. The weights, including coil, slot  
To quantify the variations of performance characteristics regarding changes in design parameters, a sensitivity analysis was conducted [29]. The influence of the design parameters on weight W and active power P was evaluated using a stepwise regression, in which the overall regression correlation coefficient, such as the R\({}^{2}\)-value, was up to 0.9. The results of the sensitivity analysis are shown in Figure 10. The source term indicated the main or interaction effect, and the LogWorth statistic value (defined as \(-\log(\text{p}-\text{value})\)) was used to measure the level at which the observed between the average criterion values was statistically significant. The LogWorth value greater than 2, indicated by the blue dashed line in these figures, was considered statistically significant for inclusion of a factor at a 1% significance level [30]. Among the design parameters, there was high sensitivity on both the active power and the weight, such as the inner diameter D\({}_{1}\), the length of the stator I and slot height h\({}_{2}\). These parameters were related to the size of the generator and the effects on electromagnetic properties, such as the magnetic flux excited permanent magnets, permeance coefficient and electromotive force [25]. The slot opening b\({}_{1}\) was related to the fundamental frequency of magnetic flux, which showed an effect on the efficiency of the generator, such as losses caused by using ferromagnetic cores for the slotted stator [25]. However, this parameter provided relatively lower sensitivity on the active power and weight. The number of winding turns per phase N\({}_{\text{ph}}\) also showed relatively lower sensitivity on both the active power and weight.