mathematics-11-01796 (1)

Mathematics 2023, 11, 1796 6 of 32
power exchange between areas would be swiftly recovered to its intended value. A timedomain
objective function is adjusted utilizing integral criteria as follows:
OF = ITAE =
∫t sim
0
(|∆ f 1 | + |∆ f 2 | + |∆P TIE |) × t × dt (8)
Furthermore, OF can be readily upgraded to minimize the peak overshoots of frequency
fluctuations across each area and in tie-line power transmission. This adaptation
evolves due to attaining a suitable damping ratio to offer a specific degree of stability [43].
The problem constraints are the boundaries of the controller component settings. Thus, the
design task could be represented as the following optimization aspect.
Subject to
For the proposed cascaded PD-PI controller,
Min OF (9)
Kp1 Area,Min ≤ Kp1 Area ≤ Kp1 Area,Max , Area = 1 or 2 (10)
Kd Area,Min ≤ Kd Area ≤ Kd Area,Max , Area = 1 or 2 (11)
Kp2 Area,Min ≤ Kp2 Area ≤ Kp2 Area,Max , Area = 1 or 2 (12)
Ki Area,Min ≤ Ki Area ≤ Ki Area,Max , Area = 1 or 2 (13)
n Area,Min ≤ n Area ≤ n Area,Max , Area = 1 or 2 (14)
The subscripts “min” and “max” represent each region’s lowest and highest quantities
of each control variable. The comparative amounts are set to be 0 and 3, and the filter
parameter n is between 0 and 500 [43].
3. Enhanced Slime Mold Optimization Algorithm
The SMOA offers a unique computing approach that uses dynamic weighting to mimic
the mechanisms that cause positive and negative reactions in the slime mold propagating
waves to form the optimum path for attaching food [27,34]. The SMOA population is
initially generated in the space with dimension (d):
V k (0) = V min + rand(0, 1)·[V max − V min ] k = 1 : NK (15)
where V min and V max represent the minimum and maximum bounds of everyone’s control
variable, and NK is the number of individuals in the population.
Considering that slime mold could follow food based on the fragrance in the air, this
behavior may be expressed as follows:
{
Vb (t) + υ
V k (t + 1) =
1
× (W × V r1 (t) − V r2 (t)) Pv > r
υ 2
× V k (t)
Pv ≤ r
k = 1 : NK (16)
where t represents the current iteration, V k represents the slime mold position, V b represents
the place having the highest smell concentrations, and V r1 and V r2 represent two options
picked randomly within the population. The slime mold selection behavior is represented
by two components, υ 1 , and υ 2 , where υ 2 decreases linearly from 1 to 0. W is the searching