**2.1. Why DOE?**

Compared to one-factor-at-a-time experiments, i.e. only one factor is changed at a time while all the other factors remain constant, the DOE technique is much more efficient and reliable. Though, the one-factor-at-a-time experiments are easy to understand, they do not tell how a factor affects a product or process in the presence of other factors (ReliaSoft Corporation, 2008). If the effect of a factor is altered, due to the presence of one or more other factors, we say that there is an interaction between these factors. Usually the interactions' effects are more influential than the effect of individual factors (ReliaSoft Corporation, 2008). This is because the actual environment of the product or process comprises the presence of many factors together instead of isolated occurrences of each factor at different times.

The DOE methodology ensures that all factors and their interactions are systematically investigated. Therefore, information obtained from a DOE analysis is much more reliable and comprehensive than results from the one-factor-at-a-time experiments that ignore interactions between factors and, therefore, may lead to wrong conclusions (ReliaSoft Corporation, 2008).

Let's assume, for instance, that we want to optimize an induction motor taking into account, for simplicity, only two factors: the length and the external radius. Hence, the length is the first factor and is denoted by �� while the external radius is the second factor and it is denoted by ��. Each factor can take several values between two limits, i.e. ������� ������ and ������� ������. We desire to study the influence of each of these factors on the system response or output for example the torque called Y. The classical or traditional approach consists of studying the two factors �� and���, separately. First we put �� at the average level ��������� and study the response of the system when �� varies between �����and ����� using for example 4 steps (experiments) as shown in Fig. 1. Similarly, we repeat the same procedure to study the effect of ��. Accordingly, the total number of tests is 8. However, we should ask a paramount question here, are these 8 experiments sufficient to have a good knowledge about the system? The simple and direct answer to this question is no. To get a better knowledge about the system, we have to mesh the validity domain of the two factors and test each node of this mesh as shown in Fig. 2. Thus, 16 experiments are needed for this investigation. In this example only two factors are taken into account. Therefore, if for example 7 factors are taken into account, the number of tests to be performed rises to 4� = 16384 experiments, which is a highly time and cost consuming process.

Knowing that it is impossible to reduce the number of values for each factor to less than 2, the designer often reduces the number of factors, which leads to incertitude of results. To reduce both cost and time, the DOE is used to establish a design experiment with less number of tests. The DOE, for example, allows identifying the influence of 7 factors with 2 points per variable with only 8 or 12 tests rather than 128 tests used by the traditional method (Bouchekara, 2011; Uy & Telford, 2009).

Recently, the DOE technique has been adopted in the design and testing of various applications including automotive assembly (Altayib, 2011), computational intelligence (Garcia, 2010), bioassay robustness studies (Kutlea, 2010) and many others.

**Figure 1.** Traditional method of experiments.

182 Induction Motors – Modelling and Control

optimization (PSO) methods.

**2. Design of Experiments (DOE)** 

statistical theories, especially with the use of DOE.

in Section 5.

capabilities.

**2.1. Why DOE?** 

factor at different times.

design method for induction motors using design of experiments (DOE) and particle swarm

The outline of this paper is as follows. The current section is the introduction. Section 2 introduces and explains the DOE method. Section 3 gives an overview of the PSO method. In Section 4 the application of the DOE and PSO to optimize induction motors is explained and its results are also presented and discussed in detail. Finally, the conclusions are drawn

With modern technological advances, the design and optimization of induction motors or any other electromechanical devices are becoming exceedingly complicated. As the cost of experimentation rises rapidly it is becoming impossible for the analyst, who is already constrained by resources and time, to investigate the numerous factors that affect these complex processes using trial and error methods (ReliaSoft Corporation, 2008). Computer simulations can solve partially this issue. Rather than building actual prototypes engineers and analysts can build computer simulation prototypes. However, the process of building, verifying, and validating induction motor simulation model can be arduous, but once completed, it can be utilized to explore different aspects of the modeled machine. Moreover, many simulation practitioners could obtain more information from their analysis if they use

In this section the DOE method is explained in order to make its use in this chapter understandable. The aim here is not to explain the whole method in detail (with all the mathematical developments behind), but to present the basics to demonstrate its interesting

Compared to one-factor-at-a-time experiments, i.e. only one factor is changed at a time while all the other factors remain constant, the DOE technique is much more efficient and reliable. Though, the one-factor-at-a-time experiments are easy to understand, they do not tell how a factor affects a product or process in the presence of other factors (ReliaSoft Corporation, 2008). If the effect of a factor is altered, due to the presence of one or more other factors, we say that there is an interaction between these factors. Usually the interactions' effects are more influential than the effect of individual factors (ReliaSoft Corporation, 2008). This is because the actual environment of the product or process comprises the presence of many factors together instead of isolated occurrences of each

The DOE methodology ensures that all factors and their interactions are systematically investigated. Therefore, information obtained from a DOE analysis is much more reliable and comprehensive than results from the one-factor-at-a-time experiments that ignore

Optimization of Induction Motors Using Design of Experiments and Particle Swarm Optimization 185

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

> **Run Factor** �� **Factor** �� **Response** � 1 -1 -1 �� 2 -1 +1 �� 3 +1 -1 �� 4 +1 +1 ��

**Figure 3.** Strategy of experimentation; points corresponding to nodes in the mesh of the experimental

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

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

of the experimental field where points correspond to nodes.

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

field for a full factorial design for 2 factors with 2 levels.

**2.5. Fractional factorial design** 

mesh?

**2.4. Full factorial design** 

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

### **2.2. Methodology**

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 Corporation, 2008).

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 interactions are not considered.

### **2.3. Mathematical concept**

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 approximated by an algebraic model given in the following equation:

$$\mathbf{x} = a\_0 + a\_1 \mathbf{x}\_1 + a\_2 \mathbf{x}\_2 + \dots + a\_k \mathbf{x}\_k + \dots + a\_1 \mathbf{x}\_1 \mathbf{x}\_2 + \dots \mathbf{a}\_1 \mathbf{x}\_1 \mathbf{x}\_k + a\_{1\dots k} \mathbf{x}\_{1\dots k} \tag{1}$$

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