Page 325 - Decision Making Applications in Modern Power Systems

286 Decision Making Applications in Modern Power Systems

Taking in the consideration that m is the smallest integer that is larger
than α, the fractional derivative is calculated based on Caputo definition as
follows [25]:

8 m
d
>
> α 5 m
> m
dt
>
<
α
a D f χ ðtÞ 5 1 ð t m ð11:15Þ
t
> D f χ tðÞ
> m 2 1!α!m
> α2m11
ð
Γ m 2 αÞ
>
: a t2τð Þ
where
ð N
Γ m 2 αÞ 5 t m2α21 Uexp 2tÞdt ð11:16Þ
ð
ð
0
To transform the computation from the time domain to the frequency
domain, the Laplace transformation is used. In the fractional calculus the
Laplace transform is given by the following equation [25]:
m21
α α X α2k21
‘ a D f χ ðtÞ 5 s FðsÞ 2 s fð0Þ ð11:17Þ
t
k50
To implement and simulate the fractional-order calculus, the Laplace
operator of the fractional order is approximated with integer-order transfer
functions. The most known method to approximate fractional order to integer
order is Oustaloup’s method [8,9,25].
11.5.2 Load-frequency control and automatic generation control
based on fractional calculus
Frequency control in power system consists of three subcontrol levels: (1)
PFC that tries to stop the frequency decline before triggering the under fre-
quency load shedding, (2) secondary frequency control known also as LFC,
which aims to mitigate the frequency deviation through a suitable controller,
and (3) tertiary frequency control, which aims to redispatch the generation
units in order to achieve the most economic operation of the system [25].
In this section, we provide an overview of LFC in the deregulated power
system modeling, which is given in Section 11.5.2.1, while a procedure of
LFC controller design based on the fractional calculus is introduced in
Section 11.5.2.2.

11.5.2.1 Load-frequency control under the deregulation
environment
The power system face is changing by its transition from vertically integra-
tion utility (VIU) structure to deregulated one. In the first structure, that is,