**2.4. Full factorial design**

184 Induction Motors – Modelling and Control

**Figure 2.** One experiment at each node of the mesh.

The design and analysis of experiments revolves around the understanding of the effects of different variables on other variable(s). The dependent variable, in the context of DOE, is called the response, and the independent variables are called factors. Experiments are run at different values of the factors, called levels. Each run of an experiment involves a combination of levels of the investigated factors. The number of runs of an experiment is determined by the number of levels being investigated in the experiment (ReliaSoft

For example, if an experiment involving two factors is to be performed, with the first factor having �� levels and the second having �� levels, then �� � �� combinations can possibly be run, and the experiment is an �� � �� factorial design. If all �� � �� combinations are run, then the experiment is a full factorial. If only some of the �� � �� combinations are run, then the experiment is a fractional factorial. Therefore, in full factorial experiments, all factors and their interactions are investigated, whereas in fractional factorial experiments, certain

Assume that � is the response of an experiment and ���� ��� �������} are � factors acting on this experiment where each factor has two levels of variation ��� and ���. The value of �, is

where �� are coefficients which represent the effect of factors and their interactions on the

���� � ���� � ���� ������� ��������� �������� � �������� (1)

approximated by an algebraic model given in the following equation:

**2.2. Methodology** 

Corporation, 2008).

interactions are not considered.

**2.3. Mathematical concept** 

response of the experiment.

As mentioned above, the study of full factorial design consists of exploring all possible combinations of the factors considered in the experiment (Kleijnen et al., 2005). Note that the design �� means that this experiment concerns a system with � factors with � levels. Usually, two levels of the �'s are used. The use of only two levels implies that the effects are monotonic on the response variable, but not necessarily linear (Uy & Telford, 2009). For each factor, the two levels are denoted using the "rating Yates" notation by -1 and +1 respectively to represent the low and the high levels of each factor. Hence, the number of experiments carried out by a full factorial design for *k* factors with 2 levels is ����. For example, Table 1 shows the design matrix of a full factorial design for 2 factors while, Fig. 3 shows the mesh of the experimental field where points correspond to nodes.


**Table 1.** Design Matrix for a full factorial design for 2 factors with 2 levels.

**Figure 3.** Strategy of experimentation; points corresponding to nodes in the mesh of the experimental field for a full factorial design for 2 factors with 2 levels.

## **2.5. Fractional factorial design**

The advantage of full factorial designs, is their ability to estimate not only the main effects of factors, but also all their interactions, i.e. two by two, three by three, up to the interaction involving all *k* factors. However, when the number of factors increases, the use of such design leads to a prohibitive number of experiments. The question to be asked here is: is it necessary to perform all experiments of the full factorial design to estimate the system's response? In other words, is it necessary to conduct a test at each node of the mesh?

It is not necessary to identify the effect of all interactions because the interactions of order ≥ 2 (like ������) are usually negligible. Therefore, certain runs specified by the full




$$a\_0 = \overline{\mathbf{y}} = \frac{1}{n} \sum\_{l=1}^{n} \mathbf{y}\_l \tag{2}$$

$$a\_{\rangle} = e\_{a\_{\rangle}} = \mathbf{y}\_{\mathbf{x}\_{\rangle}}^{+} - a\_0 \tag{3}$$

$$\mathbf{y}\_{\mathbf{x}\_{l}}^{+} = \frac{1}{n^{+}} \sum\_{l=1}^{n} \mathbf{y}\_{l}^{+} \tag{4}$$

$$\mathcal{C}\_{a\_{j}} = \frac{SCE\{a\_{j}\}}{SCE\{\mathbf{y}\}} \text{ [\%]} \tag{5}$$

$$SCE(\mathbf{y}) = \sum\_{l=1}^{n} (\mathbf{y}\_l - \overline{\mathbf{y}})^2 \tag{6}$$

$$SCE\{a\_j\} = \frac{n}{s} \sum\_{j=1}^{s} \left(e\_{a\_j}\right)^2\tag{7}$$


$$v\_l(t+1) = v\_l(t) + c\_1 \times rand(\,) \times \left(p\_l^{\text{best}} - p\_l(t)\right) + c\_2 \times rand(\,) \times \left(p\_{\text{gbest}} - p\_l(t)\right) \tag{8}$$

$$p\_l(t+1) = p\_l(t) + \nu\_l(t) \tag{9}$$

Algorithm 1: Pseudocode for PSO (Brownlee, 2011).

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 191

problem domain. Alternatively, a wrapping strategy may be used at the edge of the domain creating a loop, torrid or related geometrical structures at the chosen

Induction motors with power below 100 kW (Fig. 5) constitute a sizable portion of the global electric motor markets (Boldea & Nasar, 2002). The induction motor design optimization is a nature mixture of art and science. Detailed theory of design is not given in this chapter. Here we present what may constitute the main steps of the design methodology. For further information, see (Vogt, 1988; Boldea & Nasar, 2002; Murthy, 2008). The suitability of the DOE and the PSO techniques in induction motor design optimization will be demonstrated

An inertia coefficient can be introduced to limit the change in velocity.

**Figure 5.** Low power 3 phase induction motor with cage rotor (Boldea & Nasar, 2002).

The main steps in induction motor design optimization are shown in Fig. 6.

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

**4. Induction motor design: An optimization problem** 

dimensionality.

in this section.

**4.1. The algorithm** 

**Step (1): Initialization** 

defined.

According to (Brownlee, 2011):


problem domain. Alternatively, a wrapping strategy may be used at the edge of the domain creating a loop, torrid or related geometrical structures at the chosen dimensionality.

An inertia coefficient can be introduced to limit the change in velocity.
