*4.4.1. Results*

Using two-level full factorial design needs 26=64 runs (simulations) to evaluate objective functions. However, using a 26-2 fractional factorial design will significantly reduce the number of runs from 64 to 16. The 26-2 design matrix and the simulation results obtained for this design are given in Table 6. This design has been generated using Box generators given in Table 7. The choice of a 26-2 means that we have a 2 levels design with 6 factors where 2 of these factors are generated using the other 4 factors as shown in Table 7. Thus:


The contributions of obtained contrasts are given in Table 8. It shows in its first column contrasts and in the other columns their contribution or influences on objective functions. Keep in mind that a contribution is significant if it is higher than 5% and high order interactions (higher than 2) are considered negligible while only interactions of significant parameters are also significant.


**Table 6.** Design matrix generated by the 26-2 Box-Wilson fractional factorial design and the simulation results.



parameters are also significant.

**Cdb [A/mm2]** 

**[A/mm2]** 

**N P CDSW**

results.

*4.4.1. Results* 

Since six parameters define the shape of the motor, it is advisable to determine the effect of each parameter on the objective functions. Thus, it is very important to provide proper parameter ranges. The considered parameters are listed in Table 5. There are two types of

Using two-level full factorial design needs 26=64 runs (simulations) to evaluate objective functions. However, using a 26-2 fractional factorial design will significantly reduce the number of runs from 64 to 16. The 26-2 design matrix and the simulation results obtained for this design are given in Table 6. This design has been generated using Box generators given in Table 7. The choice of a 26-2 means that we have a 2 levels design with 6 factors where 2 of

The contributions of obtained contrasts are given in Table 8. It shows in its first column contrasts and in the other columns their contribution or influences on objective functions. Keep in mind that a contribution is significant if it is higher than 5% and high order interactions (higher than 2) are considered negligible while only interactions of significant

4 3 4 3 1 1 86.09 16.94 69.25 0.30 0.07 4 3 4 4 1 2 88.98 8.07 55.06 0.29 0.42 4 3 6 3 2 2 89.40 7.02 50.15 0.27 0.37 4 3 6 4 2 1 88.46 8.20 55.30 0.29 0.52 4 5 4 3 2 2 89.30 5.94 54.81 0.26 0.30 4 5 4 4 2 1 88.69 6.55 58.44 0.28 0.48 4 5 6 3 1 1 86.34 10.95 69.06 0.28 0.15 4 5 6 4 1 2 88.33 6.17 57.99 0.28 0.70 6 3 4 3 2 1 87.73 9.58 60.22 0.30 0.22 6 3 4 4 2 2 89.35 6.26 51.54 0.39 0.94 6 3 6 3 1 2 87.29 9.04 59.93 0.30 0.32 6 3 6 4 1 1 85.96 11.06 69.10 0.35 0.36 6 5 4 3 1 2 87.38 7.15 64.56 0.29 0.31 6 5 4 4 1 1 86.49 8.19 70.80 0.35 0.42 6 5 6 3 2 1 86.78 7.18 65.09 0.29 0.38 6 5 6 4 2 2 87.99 5.18 58.95 0.40 1.30 **Table 6.** Design matrix generated by the 26-2 Box-Wilson fractional factorial design and the simulation

**Zsw EFF Kg/kW Tr**

**[]** 

**I0/I Tst** 

**[pu]** 

these factors are generated using the other 4 factors as shown in Table 7. Thus:

 The factor (5) will be generated using the product of factors (1), (2) & (3). The factor (6) will be generated using the product of factors (2), (3) & (4).

> **Spp Tstrip [mm]**

parameters; continuous parameters and discrete parameters.


**Table 8.** Contrasts and contribution obtained.

The application of DOE identifies the effect of each parameter on each objective function. We can notice that for the efficiency Zsw, Tsrip, and P are the most significant factors with respectively 38% 34% and 13% of contribution on the objective function. Moreover, Fig. 7 gives more details. When P is low the efficiency is high and vice versa when P is high.

**Figure 7.** Plot of effects for the efficiency.

Contrariwise, when Tstrip and Zsw are low the efficiency is low, while it is high when Tstrip and Zsw are high.

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 201

is high when they are high. Inversely, for Tstrip and Zsw the temperature rise is low when

For the objective function I0/I the significant parameters are P (45%) and Spp (25)%. Furthermore, there is a significant interaction between P and Spp included in the contrast 'P Spp + Tstrip Zsw'. Fig. 10 shows that I0/I is low when each parameter is low and vise versa.

Finally, for the starting torque the most significant parameters are given in this order: Spp (4%), Zsw (19%), Tstrip (14%), P (7%) and Cdb (5%). From Fig. 11 we can notice that when

P CDSW cdb Spp Tstrip Zsw

Factors

P CDSW cdb Spp Tstrip Zsw

Factors

they are high.

**Figure 10.** Plot of effects for I0/I.

0.27

0.28

0.29

0.3

0.31

Response

0.32

0.33

0.34

**Figure 11.** Plot of effects for starting torque.

0.3

0.4

0.5

Response

0.6

For the objective function kg/kW the most important parameters are respectively Zsw (29%), Tstrip (24%), CDSW (18%) and Spp (10%). Fig. 8 shows that when each one of these parameters is low the kg/kW is high and inversely when they are high. Furthermore, for this objective function there is a significant interaction between some factors 'P Spp + Tstrip Zsw' (5%). Note that we have isolated all of the main effects from every 2-factors interaction. The two largest effects are Zsw and Tstrip, hence it seems reasonable to attribute this to the Tstrip Zsw interaction.

**Figure 8.** Plot of effects for kg/kW.

Concerning the temperature rise we can observe that, Zsw (39%), Tstrip (36%), P(9%) and CDSW (8%) are the most significant parameters. On the contrary, no significant interaction is discerned. Fig. 9 shows that the temperature rise is low when P and CDSW are low and it

**Figure 9.** Plot of effects for temperature rise.

is high when they are high. Inversely, for Tstrip and Zsw the temperature rise is low when they are high.

For the objective function I0/I the significant parameters are P (45%) and Spp (25)%. Furthermore, there is a significant interaction between P and Spp included in the contrast 'P Spp + Tstrip Zsw'. Fig. 10 shows that I0/I is low when each parameter is low and vise versa.

**Figure 10.** Plot of effects for I0/I.

200 Induction Motors – Modelling and Control

Tstrip and Zsw are high.

Tstrip Zsw interaction.

**Figure 8.** Plot of effects for kg/kW.

6.5

7

7.5

8

8.5

Response

9

9.5

10

**Figure 9.** Plot of effects for temperature rise.

62

64

66

68

Response

70

72

Contrariwise, when Tstrip and Zsw are low the efficiency is low, while it is high when

For the objective function kg/kW the most important parameters are respectively Zsw (29%), Tstrip (24%), CDSW (18%) and Spp (10%). Fig. 8 shows that when each one of these parameters is low the kg/kW is high and inversely when they are high. Furthermore, for this objective function there is a significant interaction between some factors 'P Spp + Tstrip Zsw' (5%). Note that we have isolated all of the main effects from every 2-factors interaction. The two largest effects are Zsw and Tstrip, hence it seems reasonable to attribute this to the

Concerning the temperature rise we can observe that, Zsw (39%), Tstrip (36%), P(9%) and CDSW (8%) are the most significant parameters. On the contrary, no significant interaction is discerned. Fig. 9 shows that the temperature rise is low when P and CDSW are low and it

P CDSW cdb Spp Tstrip Zsw

Factors

P CDSW cdb Spp Tstrip Zsw

Factors

Finally, for the starting torque the most significant parameters are given in this order: Spp (4%), Zsw (19%), Tstrip (14%), P (7%) and Cdb (5%). From Fig. 11 we can notice that when

**Figure 11.** Plot of effects for starting torque.

each one of these parameters is low the starting torque is low. Likewise, when these parameters are high, the starting torque is high. Furthermore, for this objective function there is two significant interaction between some factors 'P Tstrip + CDSW Cdb + Spp Zsw' (5%) and 'P Zsw + Spp Tstrip' (5%). Note that we have isolated all of the main effects from every 2-factor interaction. For the first contrast the two largest effects are Spp and Zsw. Thus, it seems reasonable to attribute this to the Spp Zsw interaction. While, for the second contrast the two largest effects are Spp and Tstrip. Hence, it is appropriate to attribute this to the Spp Tstrip interaction.

### **4.5. Optimization**

Two optimization approaches can be achieved. The first one is to treat 1 of the 5 objective functions (defined in the Objective Function section) at a time. Thus, every time a single objective function is taken into account regardless of the 4 others. The second approach is to consider a multi objective function where the 5 objective functions are taken into account at the same time. The resulted complicated multiple-objective function can be converted into a simple and practical single-objective function scalarization. Among scalarization methods we can find the weighting method. In this method, the problem is posed as follows:

$$\mathbf{F}\_{\text{objective}} = \sum\_{l=1}^{5} \mathbf{w}\_{l} f\_{l} \tag{35}$$

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 203

The PSO algorithm is implemented to optimize the design of induction motor whose specifications are given above. The results of PSO algorithm for the optimized motor are given in the Table 9. The algorithm has returned an acceptable solution every time, which is

**Parameters EFF Kg/KW Tr I0/I Tst P** 4 4 4 4 6 **CDSW** 3 5 3 5 5 **Cdb** 4 6 4 6 6 **Spp** 4 4 4 3 4 **Tstrip** 2 2 2 2 2 **Zsw** 2 2 2 2 2 **Existing Motor** 89.7 5.36 53.8 0.27 0.5 **Optimized Motor** 90.1 5.15 48.5 0.26 1.3

**Table 9.** Optimum design results for efficiency maximization, minimization of kg/kW, minimization of

According to the results presented in Table 9, when the efficiency of the motor is considered as the objective function, we can see that it increased from 89.7 to 90.1 compared to the existing motor. We can notice also that the when Kg/KW is minimized, it reduced from 5.36 to 5.15. Moreover, the optimization process allowed to the temperature rise to decrease form 53.8 to 48.5 which is a important reduction. Likewise, the I0/I is slightly reduced from 0.27 to 0.26 when it is the objective function. Finally, Table 9, shows that the starting torque is

According to these results, we can say that PSO is suitable for motor design and can reach successful designs with better performances than the existing motor while satisfying almost

This chapter investigated the optimal design of induction motor using DOE and PSO techniques with five objective functions namely, maximization of efficiency, minimization of kg/kW, minimization of temperature rise in the stator, minimization of I0/I ratio, maximization of starting torque. It has been shown that DOE and PSO based algorithms constitute a viable and powerful tool for the optimal design of induction motor. The main objective of the DEO here is to identify the effect of each parameter on the objective functions. This is of a paramount importance mainly because of two reasons. The first one and also the obvious one is the reduction of the number of parameters to be taken into

temperature rise, minimization of the ratio I0/I and starting torque maximization.

higher for the optimized motor (1.3) compared to the existing one (0.5).

every constraint.

**5. Conclusion** 

indicated by a good value for objective with no constraint violations.

where: �� = EFF, �� = −kg ⁄ kW , �� = −Tr, �� = −I0 ⁄ I, �� = Tst and �� is a constant indicating the weight (and hence importance) assigned to ��. By giving a relatively large value to �� it is possible to favor �� over other objective functions. Note that the condition ∑ �� � ��� = 1 can be posed in Eq.(35).

Nevertheless, since the 5 functions of the multi-objective function have different ranges, for instance �� varies from 85 to 91 and �� varies from 0.07 to 1.3. Thus, the values of these functions must be normalized between 0 and 1. The minimum of a given function is equal to 0 and the maximum is equal to 1. The normalization operation is given by:

$$\text{Normalized}\_{\text{Value}} = \frac{\left\{ \text{Actual}\_{\text{Value}} - \min(f\_l) \right\}}{\max(f\_l) - \min(f\_l)} \tag{36}$$

and (35) becomes:

$$\mathbf{F}\_{\text{objective}} = \sum\_{l=1}^{5} \mathbf{w}\_{l} f\_{l \text{normalized}} \tag{37}$$

For this chapter we have chosen the first approach i.e. the single objective one.

The PSO algorithm is implemented to optimize the design of induction motor whose specifications are given above. The results of PSO algorithm for the optimized motor are given in the Table 9. The algorithm has returned an acceptable solution every time, which is indicated by a good value for objective with no constraint violations.


**Table 9.** Optimum design results for efficiency maximization, minimization of kg/kW, minimization of temperature rise, minimization of the ratio I0/I and starting torque maximization.

According to the results presented in Table 9, when the efficiency of the motor is considered as the objective function, we can see that it increased from 89.7 to 90.1 compared to the existing motor. We can notice also that the when Kg/KW is minimized, it reduced from 5.36 to 5.15. Moreover, the optimization process allowed to the temperature rise to decrease form 53.8 to 48.5 which is a important reduction. Likewise, the I0/I is slightly reduced from 0.27 to 0.26 when it is the objective function. Finally, Table 9, shows that the starting torque is higher for the optimized motor (1.3) compared to the existing one (0.5).

According to these results, we can say that PSO is suitable for motor design and can reach successful designs with better performances than the existing motor while satisfying almost every constraint.
