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Decision-making-based optimal generation-side Chapter | 11 289

11.5.3 Optimal tuning of the controller parameter
11.5.3.1 The proposed objective function
In this chapter an objective function considering the settling time (ST) and
damping of the frequency oscillations of both frequency and tie-lines power
flow is used as follows:
0 1
NA ð B NA C
X t sim B X C
J 5 ω 1 U B α i Δf i ðtÞ 1 α ij ΔP tie i2j ðtÞ UtUdt
C
B C
i51 0 @ j51 A
1
j i
1
1 ω 2 U
min 1 2 ζ ; i 5 1...n ð11:26Þ
i
0 0 11
B NA B NA CC
X
B XB CC
1 ω 3 U B B STðΔf i ðtÞÞ 1 STðΔP tie i2j ðtÞÞ CC
B B CC
j51
@ i51 @ AA
1
j i
where ST is the time at which the final value of the signal settles to less
than a specific value. The weight (ω) of each term of objective function
shows the importance of each term in the objective function.
Based on the adopted FOPID controller, the optimization problem can be
described as follows:
min Jfg
s:t:
K min # K p # K p max
p
K min # K I # K max ð11:27Þ
I I
K min # K D # K max
D D
λ min # λ # λ max
μ min # μ # μ max

11.5.3.2 Imperialist competitive algorithm based fractional-
order proportional integral derivative controller’s optimization
ICA is a sociopolitical metaheuristic, inspired by the history of colonization
and competition among imperialists, to capture more colonies. The set of
countries, which are the solutions in ICA, is partitioned to form several
empires. Each empire consists of a single Imperialist and several other weak-
er countries, called colonies [20]. Two competition mechanisms are used in
the algorithm, which are the intraempire competition and the interempire