**Table 3.**

*Levels of factors.*

#### **Figure 6.**

*Graphical representation of RSM face-centered design.*

experiment (magnet thickness: 2 mm, offset: 40 mm, embrace: 1%) in this experimental design. It is not a suitable design because the offset magnet is much larger than the thickness. Therefore, since such a magnet cannot be produced, this simulation does not yield results. At the end of Maxwell simulation, it gives an error "ARC Offset is too big." The findings of 14 experimental runs using Maxwell simulations are


#### **Table 4.** *Ansys Maxwell simulation results.*

presented in **Table 4**. The disadvantage of making genuine PMSG prototypes—which is unpredictable due to expenses—is avoided in this approach. The original uncoded factor levels are also coded by using Eq. (10) given below. The mathematical modeling will be performed in terms of both uncoded and coded factor levels. Mathematical models for uncoded factor levels display the real relationship to the readers, while the models for coded factor levels will be used in the optimization phase (the details of which will be expanded in the following paragraphs).

$$\mathbf{X}\_{codeed} = \frac{\mathbf{X}\_{unode} - ((\mathbf{X}\_{max} + \mathbf{X}\_{min})/2)}{(\mathbf{X}\_{max} - \mathbf{X}\_{min})/2} \tag{10}$$

Minitab program is used for regression modeling and significance tests. The mathematical models for the uncoded factor levels are given in Eqs. (11) and (15). **Table 5** shows the *R<sup>2</sup>* statistics associated with the regression models.


**Table 5.**

*Summary of coefficient of determination values.*

*Design Optimization of 18-Poled High-Speed Permanent Magnet Synchronous Generator DOI: http://dx.doi.org/10.5772/intechopen.106987*

*<sup>Y</sup>*^<sup>1</sup> <sup>¼</sup> <sup>81</sup>*:*<sup>0151784520298</sup> <sup>þ</sup> <sup>0</sup>*:*0227200449501287*X*<sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*090808386009271*X*<sup>2</sup> <sup>þ</sup> <sup>34</sup>*:*4411803062228*X*<sup>3</sup> � <sup>0</sup>*:*0526452802359875*X*<sup>2</sup> <sup>1</sup> � <sup>0</sup>*:*00195145280235988*X*<sup>2</sup> 2 � <sup>18</sup>*:*6931720747296*X*<sup>2</sup> <sup>3</sup> þ 0*:*000938938053097341*X*1*X*<sup>2</sup> þ 0*:*513403216743922*X*1*X*<sup>3</sup> � 0*:*0987609565950274*X*2*X*<sup>3</sup>

(11)

*<sup>Y</sup>*^<sup>2</sup> <sup>¼</sup> <sup>10</sup>*:*<sup>0924651495997</sup> <sup>þ</sup> <sup>0</sup>*:*00532434330664408*X*<sup>1</sup> � <sup>0</sup>*:*00804942407641527*X*<sup>2</sup>

$$-3.24443902233461 \text{X}\_3 + 0.00413938053097343 \text{X}\_1^2 + 0.000178893805309734 \text{X}\_2^2$$

<sup>þ</sup> <sup>1</sup>*:*77530383480827*X*<sup>2</sup> <sup>3</sup> � 0*:*0000876106194690151*X*1*X*<sup>2</sup>

� 0*:*0481433487849417*X*1*X*<sup>3</sup> þ 0*:*00866354122770053*X*2*X*<sup>3</sup>

(12)

$$\begin{aligned} \hat{Y}\_3 &= 0.717642281219273 + 0.0998102261553588 \text{X}\_1 + 0.000230039331366765 \text{X}\_2 \\ &+ 0.0165307767944939 \text{X}\_3 - 0.00900958702064896 \text{X}\_1^2 - 0.00000025959 \text{X}\_2^2 \\ &\dots \end{aligned}$$

� <sup>0</sup>*:*0168692232055067*X*<sup>2</sup> <sup>3</sup> þ 0*:*0000584070796460183*X*1*X*<sup>2</sup>

þ 0*:*00464847590953787*X*1*X*<sup>3</sup> � 0*:*000464847590953787*X*2*X*<sup>3</sup>

(13)

*<sup>Y</sup>*^ <sup>4</sup> <sup>¼</sup> <sup>22</sup>*:*<sup>825129006883</sup> � <sup>0</sup>*:*0399906588003885*X*<sup>1</sup> � <sup>0</sup>*:*112362143559489*X*<sup>2</sup> �33*:*6265606686333*X*<sup>3</sup> <sup>þ</sup> <sup>0</sup>*:*0581474926253684*X*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*00201897492625369*X*<sup>2</sup> 2 <sup>þ</sup>17*:*8853726647001*X*<sup>2</sup> <sup>3</sup> þ 0*:*00091681415929201*X*1*X*<sup>2</sup> � 0*:*603758603736482*X*1*X*<sup>3</sup> þ0*:*121209193706982*X*2*X*<sup>3</sup> (14) *<sup>Y</sup>*^<sup>5</sup> <sup>¼</sup> <sup>2449</sup>*:*<sup>75496212952</sup> � <sup>0</sup>*:*949312895068736*X*<sup>1</sup> � <sup>11</sup>*:*2799654586319*X*<sup>2</sup> � <sup>4567</sup>*:*06018645878*X*<sup>3</sup> <sup>þ</sup> <sup>6</sup>*:*74597935103238*X*<sup>2</sup> <sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*257072293510325*X*<sup>2</sup> 2 <sup>þ</sup> <sup>2496</sup>*:*87546116028*X*<sup>2</sup> <sup>3</sup> � 0*:*130220353982304*X*1*X*<sup>2</sup> � 67*:*5995269700804*X*1*X*<sup>3</sup> þ 12*:*0458653954207*X*2*X*<sup>3</sup>

(15)

The *R2* values presented in **Table 5** are very close to 100%—which means the selected design parameters (magnet thickness, offset, embrace) are sufficient to mathematically model the responses. ANOVA is used to determine the model's significance. For this purpose, P-value approach is used. The summary for the ANOVA results is presented in **Table 6**.


**Table 6.** *Summary of ANOVA results.*

ANOVA results presented in **Table 6** indicate that all the calculated p-values are less than α = 0.05 (5%)—which means each mathematical model is significant and can be used in optimization phase. The RSM face-centered design looks to accurately reflect the supplied set of alternator design parameters. **Table 7** displays the prediction performances of the mathematical models. Y^<sup>i</sup> is the Minitab predictions (expected values) while Yi is the simulation results obtained from Maxwell (observed values). The prediction error percentage is denoted by PE(%) and computed using Eq. (16):

$$PE\_i(\%) = \frac{\left| Y\_i - \hat{Y}\_i \right|}{\hat{Y}\_i} \mathbf{100} \tag{16}$$

Results provided in **Table 7** show that the regression models good fit the observed values and the PE(%) is quite low. Also the confirmation tests are performed for the mathematical models. For this purpose a new dataset that is composed of five new Maxwell simulation results—which is not used in the mathematical modeling phase previously—is used. Confirmations are presented in **Table 8**.

According to the confirmation results indicated in **Table 8**, the overall PE(%) is acceptable. The comparisons shown in **Tables 7** and **8** indicate that these numerical models can be used for optimization.

In the optimization phase, four different optimization methods (RSM, GA, PSO, and MSGO) from four different classes are tested for calculating the optimum design parameters. To establish the optimum factor levels, the optimization



*Design Optimization of 18-Poled High-Speed Permanent Magnet Synchronous Generator DOI: http://dx.doi.org/10.5772/intechopen.106987*


#### **Table 8.**

*Confirmation tests.*

algorithms will be run through these five regression models. RSM is a gradientbased method, while GA (evolutionary-based algorithm), PSO (swarm intelligence-based algorithm), and MSGO (human-based algorithm) are metaheuristic optimization methods [24]. In this study, the performance of the multiobjective optimization using meta-heuristics is done by combining all the responses in one objective function independent from their units. To do this, the response functions must be recalculated by using the coded factor levels (instead of original levels) between �1 (for minimum value for the factor level) and + 1 (for maximum value for the factor level). The regression models calculated from coded factor levels are given in Eqs. (17)–(21):

$$\begin{aligned} \hat{Y}\_{1,odd} &= 96.7492070304818 + 0.01077953364239X\_1 - 1.15129382638011X\_2 \\ &+ 1.61995398230089X\_3 - 0.210581120943952X\_1^2 - 0.780581120943952X\_2^2 \\ &- 1.1683232546706X\_3^2 + 0.0375575221238924X\_1X\_2 \\ &+ 0.25670160837196X\_1X\_3 - 0.493804782975135X\_2X\_3 \end{aligned}$$

*<sup>Y</sup>*^2,*coded* <sup>¼</sup> <sup>8</sup>*:*<sup>63435501474926</sup> <sup>þ</sup> <sup>0</sup>*:*0011593271526898*X*<sup>1</sup> <sup>þ</sup> <sup>0</sup>*:*105070831577469*X*<sup>2</sup>


(18)

$$
\hat{Y}\_{3,add} = 0.990846656833825 + 0.0647760570304818X\_1 \tag{19}
$$

þ 0*:*000223942969518185*X*<sup>2</sup> þ 0*:*000130973451327435*X*<sup>3</sup>

```
� 0:0360383480825959X2
                       1 � 0:00103834808259587X2
                                                2
```

$$-0.00105432645034415X\_{3}^{2} + 0.00233628318584071X\_{1}X\_{2}$$

$$1 + 0.00232423795476892X\_1X\_3 - 0.00232423795476893X\_2X\_3$$

$$\begin{aligned} \dot{Y}\_{4,odd} &= 7.07668220255654 - 0.0185667748279251X\_1 + 1.25942010816126X\_2 \\ &- 1.6973380539735X\_3 + 0.323899705047X\_1^2 + 0.807589970501475X\_2^2 \\ &+ 1.11783579154378X\_3^2 + 0.03666725663716815X\_1X\_2 \\ &- 0.30187930186824X\_1X\_3 + 0.6060485968534907X\_2X\_3 \\\\ \dot{Y}\_{5,odd} &= 377.792067158309 - 0.571060788032038X\_1 + 150.328878248349X\_2 \\ &- 212.806948672566X\_3 + 26.9839174041298X\_1^2 + 102.82891740413X\_2^2 \\ &+ 156.054716322517X\_3^3 - 52.088141592920X\_1X\_2 \\ &- 33.7997634850401X\_1X\_3 + 60.229326971036X\_2X\_3 \\ &(21) \end{aligned}$$

The objective function is given in Eq. (22) and Eq. (23). The aim is to maximize the efficiency (Y1), while holding the air-gap flux density (Y3) at 1 Tesla and minimizing the rest of the responses (Y2, Y4, Y5).

$$\mathbf{Z} = \left| \left( Y\_{1, \text{added}} / \max(Y\_{i1}) \right) \right| - \left| \left( Y\_{2, \text{added}} / \max(Y\_{i2}) \right) \right| \tag{22}$$

$$- \left| \left( Y\_{3, \text{target}} / \max(Y\_{i3}) \right) - \left( Y\_{3, \text{odd}} / \max(Y\_{i3}) \right) \right| - \left| \left( Y\_{4, \text{odd}} / \max(Y\_{i4}) \right) \right| $$

$$- \left| \left( Y\_{5, \text{odd}} / \max(Y\_{i5}) \right) \right| $$

Min Z s.t. X1 ∈[�1,1]; X2∈[�1,1]; X3∈[�1,1]

Note that the Y3,target ¼ 1 Tesla in the equation of Z. In addition; max Yð Þi1 , max Yð Þ i2 , max Yð Þ i3 , max Yð Þ i4 , and max Yð Þi5 are the maximum observed response values presented in **Table 4** (which are 98.02, 9.06, 1.02, 11.8, and 982.58 for this problem, respectively). If the readers would like to use the Matlab codes referred in the reference [28] for MSGO, note that the signs of the each term are the exact opposite (since the codes in the reference are coded according to maximization problems). Then the *Z* function set in the Matlab code is given in Eq. (23):

$$\begin{split} Z &= -|(Y\_{1,\text{odd}}/\text{98.02})| + |(Y\_{2,\text{odd}}/\text{9.06})| + |(\text{1/1.02}) - (Y\_{3,\text{odd}}/\text{1.02})| + |(Y\_{4,\text{odd}}/\text{11.8})| \\ &+ |(Y\_{5,\text{odd}}/\text{982.58})| \end{split}$$

(23)

MSGO, PSO, GA, and RSM are run through these mathematical models to perform multi-objective optimization. **Table 9** summarizes the optimized factor levels and the calculated CPU times (at a PC: Intel i5 4GB RAM), for each method. In this table, *nPop* and *MaxIt* represent the population size and maximum number of iterations, respectively. For MSGO, c and SAP are set as 0.2 and 0.7, respectively. In PSO, the parameters of the algorithm are set as: w = 1, wdamp = 0.99, c1 = 1.5, c2 = 2.0. In the GA, we use the crossover rate = 0.50 and the mutation rate = 0.20. The optimization results for these optimization methods are presented in **Table 10**.

Results presented in **Table 10** indicate that the meta-heuristics superiors RSM with a quite bit difference. When compared among themselves, MSGO and PSO together give better results than GA. The MSGO and PSO give the same optimization results. So

*Design Optimization of 18-Poled High-Speed Permanent Magnet Synchronous Generator DOI: http://dx.doi.org/10.5772/intechopen.106987*


#### **Table 9.**

*Optimized factor levels for each method.*


#### **Table 10.**

*Summary of the optimization results.*

the optimized factor levels of MSGO and also same as PSO are used for designing the optimized design. The optimum factor levels are calculated as: magnet thickness: 5.48 mm, offset: 0 mm, embrace: 1%. However, confirmation results of these four methods are given together in **Table 11**. The observed responses and the fitted responses are presented in **Table 11**. Also the PE (%) values are calculated for each response in terms of the methods.

The optimized PMSG's magnetic flux distribution and the voltage graphs are presented in **Figures 7** and **8**, respectively. The graphs for flux linkage, power, and torque are given in **Figures 9**, **10**, and **11**, respectively.

In order to obtain the desired responses, the embrace must be maximum. Also the magnet thickness must be bigger than 4 mm. For this sample PMSG structure in the article, the results showed that offset has no discernible effect on responses (causes little changes).

As previously stated, this issue is only relevant to the PMSG in this case study. Additional optimization methods can be used to expand on these findings and discussions such as bat algorithm (BA) [29], grew wolf optimizer (GWO) [30], whale


**Table 11.** *Confirmations for the optimized factor levels.*

**Figure 7.** *Magnetic flux density distribution of the optimized PMSG.*

**Figure 8.** *Voltage graph of the optimized PMSG.*

**Figure 9.** *Flux linkage of the optimized PMSG.*

*Design Optimization of 18-Poled High-Speed Permanent Magnet Synchronous Generator DOI: http://dx.doi.org/10.5772/intechopen.106987*

**Figure 10.** *Power graph of the optimized PMSG.*

**Figure 11.**

*Torque graph of the optimized PMSG.*

optimization (WOA) [31], grasshopper optimization algorithm (GOA) [32] etc., in the future research studies.
