Soft Computing Techniques for Optimal Capacitor Placement - Complex System Modelling and Control Through Intelligent Soft Computations - page 602

(a) Algorithm

The position of the particle is continuously updated, based upon the velocity of
the particle. The velocity of the particle is calculated based on

! i ð V ! ð ! i ! i ð t Þ¼ t 1 Þþu 1 rand 1 t 1 Þ

ð 20 Þ

! g ! ð

þu 2 rand 2

Þ

t 1

where,
φ 1, φ 2

are two positive numbers

rand 1, rand 2 two random numbers with uniform distribution in the range [0,1]

V i (t) velocity of particle i at any instant t

x i (t) position of particle i at any instant t

p i local best of the particle

p g global best of the particle

Based on the velocity calculated, the updated position is given by

! i ð

! i ð ! i ð t Þ¼ t 1 Þþ t Þ ð 21 Þ

From ( 20 ) and ( 21 ), it can be clearly seen that the information available for each
particle is based on its own experience, and the knowledge of the performance of
other individuals is available in its neighbourhood.
The fl flow chart of the complete algorithm can be given in Fig. 8 .

Nelder-Mead Particle Swarm Optimization (NM-PSO)

The hybrid NM-PSO algorithm, as proposed by Fan and Zahara (Zahara and Kao
2009 ), is based on Nelder-Mead (NM) simplex search method and PSO for opti-
mization. Nelder and Mead proposed a simple local direct search technique, which
does not require any derivative for

finding solution of any function (Nelder and
Mead 1965 ). The PSO as proposed is a global search technique, but it is limited by
high computational cost of the slow convergence rate.
The slow convergence rate of PSO as compared to local search techniques like
NM is due to improper utilization of local information to determine a most
promising search direction. To overcome this slow convergence, the PSO is
combined with local simplex search technique, in a way that both the algorithms
enjoy merits of each other. In NM-PSO, the PSO prevents the hybrid approach from
being trapped in local optima, whereas the convergence rate is increased by NM
algorithm. In other words, the PSO focuses on “ exploration ” and the simplex

method, i.e., NM algorithm, focuses on Out of two components of

NM-PSO, viz., NM and PSO, PSO is described in the previous sub-section, while
the NM is discussed below.

“ exploitation. ”