Page 294 - Decision Making Applications in Modern Power Systems

254 Decision Making Applications in Modern Power Systems

and lbest (best local) models. These models differ in the manner that they
interact between a determined particle and its particles set. In this study the
gbest model is considered; thus the particle with the best suitability of entire
population, ^ y, is determined by the following equation, being iA1; ... ; s.

^ ytðÞA y o tðÞ; y 1 tðÞ; ... ; y s tðÞ9 f ^ ytðÞð Þ 5 min fy o tðÞÞ; fy 1 tðÞÞ; ... ; fy s tðÞފ
ð
ð
½
ð
ð10:13Þ
Step 3: For each particle i in s,
1. Update the velocity (10.14):

v i;k ðt 1 1Þ 5 wUv i;k ðtÞ 1 c 1 Ur 1;k ðtÞU y i;k ðtÞ 2 x i;k ðtÞ
ð10:14Þ
1c 2 Ur 2;k ðtÞU ^ y ðtÞ 2 x i;k ðtÞ
k
where
1. r 1 and r 2 are two independent random sequences, r 1 B 0; 1ð Þ and
r 2 B 0; 1ð Þ, and contribute to the nature of stochastic algorithm.
2. c 1 and c 2 are constants, called acceleration coefficients, and influence the
maximum step size for a particle in the same iteration, c 1 . 0 and c 2 # 2,
where c 1 is the coefficient that regulates the maximum step toward y i and
c 2 toward ^ y.
3. w is the inertia weight and was introduced in (10.14) by [18] as one of
the modifications in the original algorithm. In general the value of w is
varied linearly at each iteration according to the following equation,
being iter max the iterations maximum number, iter denotes each iteration,
w max the maximum inertia weight, and w min the minimum inertia weight.
w max 2 w min
w 5 w max 2 iter ð10:15Þ
iter max
4. To the system not extrapolating the search space; the particle velocity is
limited by (10.16). If the search space is defined by interval
½ 2 x max ; x max Š; the value v max [19] is calculated by (10.17):

v i 5 v max ; &v i . v max ð10:16Þ
2v max ; &v i ,2 v max
v max 5 kx max ;0:1 , k # 1:0 ð10:17Þ

2. Update the position (10.18):

x i ðt 1 1Þ 5 x i ðtÞ 1 v i ðt 1 1Þ ð10:18Þ
Step 4: If the objective is not reached, return to Step 1.