Page 564 - Mathematical Techniques of Fractional Order Systems

542 Mathematical Techniques of Fractional Order Systems

The f and l values are changed abruptly with the chaos maps over the
course of iteration, which may help the trapped grasshoppers to release them-
selves from local minima. Therefore, the chaos maps are normalized to lie in
the same ranges as follows in Eq. (18.21).

~
NormChaos 5 Ch~ aos:ðd 2 eÞ 1 e ð18:21Þ
b 2 a
~
l 5 NormChaosðiÞ ð18:22Þ
~
f 5 NormChaosðiÞ ð18:23Þ
where [ab] are the range of chaotic maps. [ed] are the range of the normali-
zation. The [ed] interval is selected as [0.3 0.7] and [1.3 1.7] for f and l,
~
respectively. NormChaos is the normalized chaos map.

18.5 SIMULATION AND RESULTS

The parameters of the commensurate and incommensurate fractional order
PMSM models are extracted using the GWO, GOA, CGWO, and CGOA
techniques. Statistical analysis is performed to compare the results of the
proposed algorithms. This analysis includes the mean value and the Standard
deviation (STD) of the identified models parameters in addition to the STD
of the fitness function (MSE). Moreover, the mean and the STD of the abso-
lute Percentage Error (APE) between the identified parameters and the
reported global values in (Li et al., 2012; Xue et al., 2015) are calculated.
Both of the STD and mean absolute percentage error (MAPE) can be calcu-
lated as follows:

Standard Deviation (STD)
N
1 X
STD 5 ZðiÞ 2 ZðiÞ ð18:24Þ
N
i51
where N is the number of runs, Z is the extracted values such as estimated
parameters or MSE or APE at each run, while Z is the mean value of
these extracted values over the total number of runs.

Mean absolute percentage error (MAPE) of the estimated parameters is
calculated as follows:
N ^
100 X YðiÞ 2 YðiÞ
MAPE 5 ð18:25Þ
N Y
i51
^
where Y is the global value of the model parameters and Y is the estimated
parameters values.