Page 552 - Mathematical Techniques of Fractional Order Systems

Parameters Identification of Fractional Order Chapter | 18 531

Chaotic Particle Swarm Optimization (Liu et al., 2005), Chaotic Bee Colony
Optimization (Alatas, 2010b), Chaotic Harmony Search (Alatas, 2010a), and
Chaotic Bat Optimization Algorithm (Rezaee Jordehi, 2015; Gandomi and
Yang, 2014).
In this chapter, two main targets are accomplished. The first one is the
utilization of commensurate and incommensurate fractional order differential
equations for modeling of the PMSM to achieve a better description of the
motor physical behavior. The second target is introducing novel developed
meta-heuristic optimization algorithms to identify the parameters of com-
mensurate and incommensurate fractional order PMSM models correspond-
ing to the chaotic behavior in the motor. The developed applied algorithms
such as Chaotic Grey Wolf Optimizer and the Chaotic Grasshopper
Optimization Algorithms, in addition to their original versions which are
called Grey Wolf Optimizer and Grasshopper Optimizer, are proposed for
the first time. The comparison between the results of the introduced devel-
oped algorithms and their original versions is carried out to recommend the
more suitable optimization technique to estimate the models parameters effi-
ciently. The final outcome clarifies that the chaotic algorithms extract the
models parameters accurately with less error and less execution time com-
pared with the original ones.
The rest of the manuscript is organized as follows: Section 18.2 presents
the PMSM motor models. The problem formulation is described in section
18.3. Section 18.4 introduces an overview for the meta-heuristic algorithm.
Simulation and results are investigated in section 18.5. Section 18.6 presents
the conclusion.

18.2 PMSM MODELS

In general, the PMSM model can be described as a system of nonlinear dif-
ferential equations (Liu et al., 2008). These equations can be established as
fractional order differential equations with noninteger derivative orders that
may have fractional values lower or greater than 1 (Li et al., 2012).

18.2.1 Fractional order PMSM
In the fractional order model of PMSM, the system of differential equations
are formulated with noninteger derivative orders. This feature enhances the
flexibility of the PMSM model design by providing extra degrees of freedom
in the model (Xue et al., 2015; Li et al., 2012). In Xue et al. (2015), the
authors propose a commensurate fractional order PMSM model where all
derivative orders have equal values. However, in Li et al. (2012), the incom-
mensurate fractional order PMSM model is introduced with the derivative
orders of different values. Generally the system of the differential equations
can be written as follows 18.1 (Li et al., 2012)