Page 415 - Decision Making Applications in Modern Power Systems

376 Decision Making Applications in Modern Power Systems

DPF limits
Its value measured at the PCC should have a lagging value between
95% and 100% according to the IEEE Standard 519. Thus a DPF con-
straint is formulated as follows:

95ðlaggingÞ # DPF %ðÞ # 100 ð14:33Þ
The thermal limit (current limit) for the IG of FSWECS
Maximum one of the rms phase currents (I Ga ; I Gb ; I Gc ) of the
FSWECSs should not exceed the rated current (I GR ):

MaxðI Ga ; I Gb ; I Gc Þ # I GR ð14:34Þ

14.3.3 Particle swarm optimization algorithm
PSO is a stochastic optimization algorithm that was developed by Dr.
Eberhart and Dr. Kennedy in 1995 [34 36]. PSO algorithm is based on bird
swarms that communicated with each other, for instance, to find food. In this
algorithm a bird is expressed as a candidate of solution, and all birds in the
group together are called “swarm.”
The best solution is found regarding fitness function output. Velocity and
position values of each agent are randomly initialized, and then they are
updated using random variables and mathematical equations in terms of
global best (gbest), position of the best fitness value in the whole process,
and the personal best value of each agent (pbest). The velocity values are
accelerated over the gbest and pbest values by using the following equations:
k11 k k k k k
v id 5 w 3 v id 1 c 1 3 U 3 ð pbest 2 x id Þ 1 c 2 3 U 3 gbest 2 x id
id d
ð14:35Þ
k11 k k11
x id 5 x id 1 v id ð14:36Þ
where d denotes a column vector in m-dimension (d 5 1; 2; ... ; m), i denotes
a row vector in the n-dimension (i 5 1; 2; ... ; n), k is the iteration number,
k
k
v id is the velocity value of ith agent at the kth iteration, x id is the current
position of ith agent at the kth iteration; pbest k is the personal best value of
id
the ith agent at the kth iteration, gbest k is the global best value of the ith
d
agent at the kth iteration, U is the random values between 0 and 1, w is the
inertia weight, c 1 and c 2 are the weighting factors, they are selected as 2 in
this study. w decreases in the interval (0.9 0.4) as given in the following
equation:
w max 2 w min
w 5 w max 2 3 iter ð14:37Þ
iter max
where w max and w min are bounds of w, iter max is the maximum iteration number,
and iter is the current iteration number.