**Step (1): Initialization**

The design process may start with design specifications and assigned values of: rated power, nominal voltage, frequency, power factor, type (squirrel Cage or slip-ring), connection (star or delta), ventilation, ducts, iron factor, insulation, curves like B/H, losses, Carter coefficient, tables like specific magnetic loading, specific electric loading, density etc. Then, design constraints for flux densities, current densities are specified. After that, the computer program is formulated with imposing max & min limits for rotor peripheral speed, length/pole pitch, stator slot-pitch, number of rotor slots. Finally, suitable values for certain parameters are assumed and objective functions are defined.

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 193

In this step the parameters to be taken into account in the optimization process are selected. The selection of parameters may be chosen by the designer or imposed by the user (for

While there are potentially many parameters (factors) that affect the performance (objective functions) of the induction motor, some parameters are more important, *viz,* have a greater impact on the performance. The DOE provides a systematic & efficient plan of experimentation to compute the effect of factors on the performance of the motor, so that several factors can be studied simultaneously (Bouchekara, 2011). As said earlier, the DOE technique is an effective tool for maximizing the amount of information obtained from a study while minimizing the amount of data to be collected (Bouchekara, 2011). The DOE technique is used here to reduce the number of parameters (screening) to be taken into account in the optimization process. This goal is achieved by identifying the effect of each parameter on the objective function to be optimized. Only significant parameters (with

Total design is split into six parts in a proper sequence as shown in Fig. 6. The sequential steps for design of each part are briefly describes in the following sub sections. For more

where: kW is the rating power, Bav is the specific magnetic loading, q is the specific electric

Then the rotor volume that is (rotor diameter D)2 × (rotor length L) is computed using the

kW

τ� × L × Bav

D�L =

ϕ =

C0 = 11 × kW × Bav × q × EFF × pf × 10�� (10)

CO × ns (11)

10� (12)

contribution higher than 5%) are considered in the optimization step.

**Step (2): Parameter selection** 

specific application for instance).

**Step (3): Parameter screening** 

**Step (4): Design** 

following formula:

details see (Murthy, 2008).

**Part I: Design of magnetic frame**

In this part the output coefficient (C0) is calculated by:

loading, EFF is the efficiency and pf is the power factor.

where: ns is the synchronous speed measured in rps.

Finally, the flux per pole is calculated by:

where: τ� is the pole pitch and its is given by:

**Figure 6.** Flowchart for computer-aided optimal design of 3-ph induction motor.

### **Step (2): Parameter selection**

192 Induction Motors – Modelling and Control

**Figure 6.** Flowchart for computer-aided optimal design of 3-ph induction motor.

In this step the parameters to be taken into account in the optimization process are selected. The selection of parameters may be chosen by the designer or imposed by the user (for specific application for instance).

### **Step (3): Parameter screening**

While there are potentially many parameters (factors) that affect the performance (objective functions) of the induction motor, some parameters are more important, *viz,* have a greater impact on the performance. The DOE provides a systematic & efficient plan of experimentation to compute the effect of factors on the performance of the motor, so that several factors can be studied simultaneously (Bouchekara, 2011). As said earlier, the DOE technique is an effective tool for maximizing the amount of information obtained from a study while minimizing the amount of data to be collected (Bouchekara, 2011). The DOE technique is used here to reduce the number of parameters (screening) to be taken into account in the optimization process. This goal is achieved by identifying the effect of each parameter on the objective function to be optimized. Only significant parameters (with contribution higher than 5%) are considered in the optimization step.

### **Step (4): Design**

Total design is split into six parts in a proper sequence as shown in Fig. 6. The sequential steps for design of each part are briefly describes in the following sub sections. For more details see (Murthy, 2008).

### **Part I: Design of magnetic frame**

In this part the output coefficient (C0) is calculated by:

$$\text{LCO} = \text{ } 11 \times \text{kW} \times \text{Bav} \times \text{q} \times \text{EFF} \times \text{pf} \times 10^{-3} \tag{10}$$

where: kW is the rating power, Bav is the specific magnetic loading, q is the specific electric loading, EFF is the efficiency and pf is the power factor.

Then the rotor volume that is (rotor diameter D)2 × (rotor length L) is computed using the following formula:

$$\text{D}^2\text{L} = \frac{\text{kW}}{\text{CO} \times \text{ns}} \tag{11}$$

where: ns is the synchronous speed measured in rps.

Finally, the flux per pole is calculated by:

$$
\phi = \frac{\tau\_{\rm p} \times L \times \text{Bav}}{10^6} \tag{12}
$$

where: τ� is the pole pitch and its is given by:

$$
\pi\_{\rm p} = \frac{\pi \times D}{P} \tag{13}
$$

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 195

ATT = ATS + ATR + ATg (23)

2 × 1.17 × kW × Tph (24)

I0 = �Iw� + Im� (25)

Total Reactance ph ⁄ = Xs + X0 + Xz (27)

R

I0 (26)

<sup>Z</sup> (28)

<sup>Z</sup> (29)

Weight of Rotor Copper (Wcur) = Lb × Sr × Ab × 8.9 × 10�� (21)

Weight of Rotor End − Rings (Weue) = π × Dme × 2 × Ae × 8.9 × 10�� (22)

where: Lb is the length of bar, Sr is the number of Rotor Slots, Ab is the rotor bar area, Ae

First, the total ampere turns (ATT) for the motor are calculated using (23). Then, the magnetizing current (Im) is calculated using (24). Finally, the no load phase current (I0) and

P × ATT

Iw

where: ATS, ATR and ATg are the total ampere turns for the stator, the rotor and the air gap

In this part the total reactance per phase, short-circuit current, and short-circuit power factor

Short Circuit Current (Isc) = Vph

Short Circuit pf =

where: Xs is the slot reactance, X0 is the overhang reactance , Xz is the zig-zag reactance, R is

In this last part of the design the performance of the induction motor are evaluated. The efficiency, the slip, the starting torque, the temperature rise and the total weight per kilo

pf0 =

the area of cross sectional of end ring and Dme is mean diameter of end-ring.

the no load power factor (pf0) are calculated using respectively (25) and (26).

Im =

**Part IV: Total ampere turns and magnetizing current** 

and Iw is the Wattful current.

**Part V: Short-circuit current calculation** 

the resistance and Z is the impedance.

watt are calculated using the following formulas:

**Part VI: Performance calculation**

are calculated using the following formulas:

### **Part II: Design of stator winding**

The first step of this part consists of calculating the size of slots using the following equations:

$$\text{Slot Width(Ws)} = \text{[Zsw} \times \text{(Tstrip} + \text{insS)} + \text{insW}] \tag{14}$$

$$\text{Slot Height (Hs)} = \text{[Zsh} \times \text{(Hstrip} + \text{insS}) + \text{Hw} + \text{HL} + \text{insH}] \tag{15}$$

where: Zsw is the width-wise number of conductors, Tstrip is the assuming thickness of strip/conductor, insS is the strip insulation thickness, insW is the width-wise insulation, Zsh is the number of strips/conductors height-wise in a slot, Hstrip is the height of the strip, HL is height of lip, Hw is the height of wedge and insH is the height-wise insulation.

Then, the copper losses and the weight of copper are calculated by:

$$\text{Copper Losses (Pcus)} = \mathbf{3} \times \mathbf{lph}^2 \times \mathbf{Rph} \tag{16}$$

$$\text{Weight of Copper (Wcus)} = \text{Lmt } \times \text{Tph} \times 3 \times \text{As} \times 8.9 \times 10^{-3} \tag{17}$$

where: Iph is the current per phase, Rph is the resistance at 20°C, Lmt is the mean length of turn, Tph represents the turns per phase and As is the area of strip/conductor.

Finally, the iron losses are calculated by multiplying the coefficient deduced from the curve giving the losses in (W/kg) in function of the flux density in (T) by the core weight.

### **Part III: Design of Squirrel Cage Rotor**

First, the air gap length is calculated by:

$$\text{Air} - \text{Gap Length (Lg)} = 0.2 + 2 \times \sqrt{\text{D} \times L \times 10^6} \tag{18}$$

Then, the rotor diameter is calculated using the following formula:

$$\text{Rotor Diameter (Dr)} = \text{D} - \text{Z} \times \text{Lg} \tag{19}$$

Finally, the copper losses and the rotor weight are calculated using equations (20), (21) and (22).

$$\begin{array}{rcl} \text{Total Rotor Copper Loss (Pour)}\\ &= \text{copper Loss in the Bars} + \text{copper Losses in the 2 End Rings} \end{array} \tag{20}$$

$$\text{Weight of Rotor COP (Wcur)} = \text{Lb} \times \text{Sr} \times \text{Ab} \times 8.9 \times 10^{-6} \tag{21}$$

$$\text{Weight of Rotor End} - \text{Rings (Weue)} = \pi \times \text{Dme} \times 2 \times \text{Ae} \times 8.9 \times 10^{-6} \tag{22}$$

where: Lb is the length of bar, Sr is the number of Rotor Slots, Ab is the rotor bar area, Ae the area of cross sectional of end ring and Dme is mean diameter of end-ring.

### **Part IV: Total ampere turns and magnetizing current**

194 Induction Motors – Modelling and Control

**Part II: Design of stator winding**

**Part III: Design of Squirrel Cage Rotor**  First, the air gap length is calculated by:

Total Rotor Copper Loss (Pcur)

equations:

insulation.

(22).

τ� <sup>=</sup> π×D

The first step of this part consists of calculating the size of slots using the following

where: Zsw is the width-wise number of conductors, Tstrip is the assuming thickness of strip/conductor, insS is the strip insulation thickness, insW is the width-wise insulation, Zsh is the number of strips/conductors height-wise in a slot, Hstrip is the height of the strip, HL is height of lip, Hw is the height of wedge and insH is the height-wise

Then, the copper losses and the weight of copper are calculated by:

Then, the rotor diameter is calculated using the following formula:

Slot Width(Ws) = [Zsw × (Tstrip + insS) + insW] (14)

Copper Losses (Pcus) = 3 × Iph� × Rph (16)

Air − Gap Length (Lg) = 0.2 + 2 × �D × L × 10� (18)

Rotor Diameter (Dr) = D − 2 × Lg (19)

= Copper Loss in the Bars + Copper Losses in the 2 End Rings (20)

Weight of Copper (Wcus) = Lmt × Tph × 3 × As × 8.9 × 10�� (17)

where: Iph is the current per phase, Rph is the resistance at 20°C, Lmt is the mean length of

Finally, the iron losses are calculated by multiplying the coefficient deduced from the curve

Finally, the copper losses and the rotor weight are calculated using equations (20), (21) and

turn, Tph represents the turns per phase and As is the area of strip/conductor.

giving the losses in (W/kg) in function of the flux density in (T) by the core weight.

Slot Height (Hs) = [Zsh × (Hstrip + insS) + Hw + HL + insH] (15)

<sup>P</sup> (13)

First, the total ampere turns (ATT) for the motor are calculated using (23). Then, the magnetizing current (Im) is calculated using (24). Finally, the no load phase current (I0) and the no load power factor (pf0) are calculated using respectively (25) and (26).

$$\text{ATT} = \text{ATS} + \text{ATR} + \text{ATg} \tag{23}$$

$$\text{Im} = \frac{\text{P} \times \text{ATT}}{2 \times 1.17 \times \text{kW} \times \text{Tph}} \tag{24}$$

$$10 = \sqrt{\text{Iw}^2 + \text{Im}^2} \tag{25}$$

$$\text{pf0} = \frac{\text{Iw}}{\text{I0}} \tag{26}$$

where: ATS, ATR and ATg are the total ampere turns for the stator, the rotor and the air gap and Iw is the Wattful current.

### **Part V: Short-circuit current calculation**

In this part the total reactance per phase, short-circuit current, and short-circuit power factor are calculated using the following formulas:

$$\text{Total Reactance/ph} = \text{Xs} + \text{X0} + \text{Xz} \tag{27}$$

$$\text{Short Circuit Current (lsc)} = \frac{\text{Vph}}{\text{Z}}\tag{28}$$

$$\text{Short Circuit pf} = \frac{\text{R}}{\text{Z}}\tag{29}$$

where: Xs is the slot reactance, X0 is the overhang reactance , Xz is the zig-zag reactance, R is the resistance and Z is the impedance.

### **Part VI: Performance calculation**

In this last part of the design the performance of the induction motor are evaluated. The efficiency, the slip, the starting torque, the temperature rise and the total weight per kilo watt are calculated using the following formulas:

$$\text{Efficiency (EFF)} = \frac{\text{kW}}{\text{KW} + \text{Total Losses}} \tag{30}$$

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 197

Value Maximum Value Type

variables (Thanga, 2008). Therefore variables selection is important in the motor design optimization. A general nonlinear programming problem can be stated in mathematical

�� is known as objective function which is to be minimized or maximized; ��'s are constants and ��'s are the variables. The following variables and constraints (Thanga, 2008) are

P Number of poles 4 6 Discrete CDSW Stator winding current density 3 [A/mm2] 5 [A/mm2] Continuous cdb Current density in rotor bar 4 [A/mm2] 6 [A/mm2] Continuous Spp Slots/pole/phase 3 4 Discrete Tstrip Stator conductor thickness 1 [mm] 2 [mm] Continuous

wise 1 2 Discrete

Five different objective functions are considered while designing the machine using

Here, the DOE is applied to analyze the objective functions. The proposed approach uses tools of the experimental design method: fractional designs, notably of Box generators to estimate the performance of the induction motor. The interest is to save calculation time and to find a near global optimum. The saving of time can be substantial because the number of

terms as follows.

��(�) 0, � = �� �� � �

*4.3.1. Variables* 

Find � = (��� �� �..��) such that ��(�) is a minimum or maximum

considered to get optimal values of objective functions.

Name Description Minimum

**Table 5.** Design optimization parameters with their domains.

optimization algorithm. The objective functions are,

1. Maximization of efficiency; F�(x) = max (EFF). 2. Minimization of kg/kW; F�(x) = min (kg/kW).

4. Minimization of I0/I ratio; F�(x) = min (I0/I). 5. Maximization of starting torque; F�(x) = max (Tst).

**4.4. Fractional 2 levels factorial design** 

simulations needed is significantly reduced.

3. Minimization of temperature rise in the stator; F�(x) = min (Tr).

The variables considered are given in Table 5.

Zsw Number of conductors width-

*4.3.2. Objective functions* 

$$\text{Slip at Full Load (SFL)} = \text{Total Rotor copper loss} \times \text{Rotor Input} \times 100\tag{31}$$

$$\text{Starting Torque (Tst)} = \left(\frac{\text{Isc}}{\text{Ir}}\right)^2 \times \text{Slip at Full Load} \tag{32}$$

$$\text{Temperature Rise (Tr)} = 0.03 \times \frac{\text{Total stator Losses}}{\text{Total Coading Area}} \tag{33}$$

$$\text{kg/kW} = \frac{\text{Total Weight}}{\text{kW}} \tag{34}$$

where: Isc is the short circuit current and Ir is the equivalent rotor current.

At the end of step (4) an automatic check is performed. If the design constraints are satisfied we move to step (5) otherwise step (4) is restarted with new values of parameters.

### **Step (5): Optimization**

In this step the motor's performances are checked and if found unsatisfactory, the process is restarted in step (4) with new values of parameters. The decision is made based on the PSO optimization method.
