IEEE Computational Intelligence Magazine - August 2021 - 55

into three phases: the gas phase, the liquid
phase, and the solid phase. The gas phase
corresponds to the previous solution set. In
the gas state, the molecules are far from one
another and widely distributed, and intense
motions and collisions occur among them.
The liquid phase represents the intermediate
solution set. In the liquid state, the distances between molecules
are considerably smaller than those in the gas state, the
distribution is relatively concentrated, and the movements are
relatively limited. Finally, the solid phase represents the later
solution set. In the solid state, the distances between molecules
are minimal, the distribution is highly concentrated,
and only a small amount of motion occurs.
In each phase, three operations are executed between the
molecules, defined as follows:
(1) Direction Vector Operator: The purpose of the direction
vector operator is to push other individuals to move
toward the best individual in the population. Let Pi
sent a member of the population, and let pbest
current best individual. Pbest
population, causing them to move toward P .best Let di
reprerepresent
the
attracts other members of the
be the
direction vector of the i-th member. The corresponding
equation is as follows:
dd .,
epoch
i
t
+1=- +cm 5))t
i
t
PP
10 PP
best
best
-
-
i
i
(5)
where t denotes the current iteration number and epoch
denotes the total number of iterations. The obtained value di
is used to calculate the velocity vector of the i-th member of
the population, as follows:
n
vd
ii
= ))c
-
/^h
low
bbm
high
m
m=1
,
n
(6)
where c is a constant positive value with a range of [0,1] and
n represents the dimensionality of each member, namely, the
number of parameters in the model. The upper and lower
bounds on the m-th parameter are bm
Next, we can employ vi
high and
update function is defined as follows:
+1
Pp vRandb b
,, m=+ )) ) a^h (7)
im
t
where Rand1
im
t
1
high
low
m - m
,
is a random number between 0 and 1 and a is a
constant positive value with a range of [0,1].
(2) Collision Operator: When two molecules approach
each other, they may collide. The collision operator emulates
this process; if two individuals have collided, their
direction vectors are exchanged, which ensures the diversity
of the population and prevents premature convergence in
the evolutionary process. Collision behavior occurs when
two individuals are sufficiently close to each other. Let r
TABLE I Parameter settings of the SMS algorithm
in different states.
STATE
GAS
DURATION c
50%
LIQUID 40%
SOLID
10%
a
0.8
0.4
0.8
0.2
0.1 0
b
[0.8,1]
[0.0,0.6]
[0.0,0.1]
PROBABILITY H
0.9
0.2
0.0
b ,m
low
respectively.
to update the position p ,i where the
P , = with probability H ,
+1
im
t
+1
Z
[
\
]]
]
im
t
,
(3) Random Behavior: Following the rules of the SMS
algorithm, during the iterative process, each individual will
likely exhibit random behavior, i.e., the elements of the individuals
will randomly change. The function for this random
behavior can be expressed as follows:
bRandb b
PH
low
m +- h
,,
,
2 ) ^ m
low
m
(10)
with probability ^1h
where
H represents the occurrence probability of the random
behavior.
The entire iterative process in the SMS algorithm is
divided into three phases by establishing different c , a , b
and H values. In accordance with [38], we adopt the parameters
presented in Table II for the three phases. The complete
SMS process is described in Algorithm 1.
III. Wind Speed Forecasting Framework
The process of applying DNR for wind speed prediction is
illustrated in Fig. 2. First, the time delay and embedding
dimensionality of the series are calculated to reconstruct the
phase space. Next, the maximum Lyapunov exponent is calculated
from the time delay and embedding dimensionality.
When this exponent exceeds zero, the wind speed series
data are regarded as a chaotic time series. Subsequently, the
high
The process of searching for the best solution in the
SMS algorithm can be expressed as a series of physical
motions among molecules, which mimic the state
transformations of matter.
denote the corresponding distance threshold, which is calculated
as follows:
n
/^h
low
bbm
r =
m=1
n
high - m
) b ,
(8)
where b is a constant positive value with a range of [0,1].
If the distance between two individuals is less than r, they
will undergo col lision, and this phenomenon can be
expressed as follows:
dd ANDd d== (9)
mi
im
.
AUGUST 2021 | IEEE COMPUTATIONAL INTELLIGENCE MAGAZINE 55
IEEE Computational Intelligence Magazine - August 2021

Table of Contents for the Digital Edition of IEEE Computational Intelligence Magazine - August 2021
