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EMP = { PMU(0,
x ),
PMU(1,
x ), ⋅⋅⋅PMU(
i,
x ) ⋅⋅⋅,
0
PMU(
N −1,
x )}
N −1
Where, PMU ( i,
) stands for the x -th PMU in
i
x i
1
i
⑾
G .
i
B. Designing the relative fitness function
1) Influence degree and influence degree coefficient
Usually, EMP that simultaneously satisfies the
three objectives does not exist. So the weights should
be assigned to the three objectives. The three weights
are expressed byW , W andW :
C
T
W
C+ WT
+ WQ
=1
⑿
According to the coding regulations, a chromosome
(an individual) is described as X = [ x0x1
⋅⋅⋅ x i
⋅⋅⋅ xN
−1]
,
which stands for an EMP. Where: x is an integer,
0 ≤ xi p m i
, and m
i
is the number of PMU in G
i
. For
each chromosome (individual), the ( C , T,
Q)
value
can be computed through ⒀ in every generation.
Then, the objective eigenvalue matrix is computed and
described as ⒁.
N−1
N−2
⎧
C( X)
= Cin
+ Ctr=
+
⎪
∑C
i(
xi
) ∑C
i,
i+
1(
xi
, xi
+ 1)
i=
0
i=
0
⎪
N−1
N−2
⎨T( X)
= Tin
+ Ttr=∑
Ti
( xi
) + ∑T
i,
i+
1(
xi
, xi
+ 1)
i=
0
i=
0
⎪
N−1
⎪Q( X)
= ∑(1-Q i(
xi
))
⎩
i=
0
⎡C(
X1)
C(
X2)
... C(
XM)
⎤
⎢
⎥
⎢
T(
X1)
T(
X2)
... T(
X )
⒁
M
⎥
⎢⎣
Q(
X ) ( ) ... ( ) ⎥
1
Q X2
Q XM
⎦
Cmax , Cmin,
Tmax,
Tmin,
Qmaxand Q min
can be computed
based on the eigenvalue matrix of a generation.
For a new generation, each individual is newly
produced. The authors define this influence as the
influence degree of the objective non-standardization,
marked as E , E and E , computed by ⒂:
Where:
C
T
Q
Q
i
⒀
α
⎧ ⎛ C ⎞
max
−Cmin
⎪E
⎜
⎟
C
= WC
⋅
⎪ ⎝ Cmax+ Cmin
⎠
⒂
⎪
α
⎛T
− ⎞
max
Tmin
⎨ E = ⋅
⎜
⎟
T
WT
⎪ ⎝Tmax
+ Tmin
⎠
α
⎪ ⎛ Q −Q
⎞
max min
⎪E
= W ⋅
⎪
⎜
⎟
Q Q
⎩ ⎝ Q + Q
max min ⎠
C − C
max min , T − T
max min and Q − Q
max min
C +C
T + T
Q + Q
max
min
respectively express the relative changing range of C, T
and Q. α is the adjusting factors. Considering that the
weights have the more direct influence on the
optimization result than the eigenvalue relative changing
range usually α is given a positive value less than 1. In
this paper, α = 0. 5 .
Then, the three influence degrees expressed by ⒂
are compared with each other, and the maximum and
minimum influence degree can be obtained, marked as
E and E
max
min
. The influence degree coefficient
e e , e , e ) is computed by ⒃.
(
c t q
max
min
max
min
⎧ E − E
C min
⎪ e =
c
⎪ E
max
− E
min
⎪ E − E
⒃
T min
⎨ e
t
=
⎪ E − E
max
min
⎪ E − E
Q min
⎪ e
q
=
⎩ E − E
max
min
2) Relative membership degree based on influence
degree coefficients
From ⑸, we can know that the less the three
objective values of an individual are, the better the
individual is. So, if Cmin ,Tmin
and Qmin
are far
to Cmax ,Tmax
and Qmax
respectively, the strong relative
membership degree of an individual respectively
corresponding to C min
, Tmin
and Q min
is defined as ⒄:
⎧
C - C ( X )
max
⎪ r1
C
( X ) =
⎪
C - C
max
min
⒄
⎪
T − T ( X )
max
⎨ r ( X ) =
1 T
⎪
T − T
max
min
⎪
Q − Q ( X )
max
⎪
r ( X ) =
1 Q
⎩
Q − Q
max
min
But, if Cmin,Tmin
and Qmin
are near
to Cmax,Tmax
and Q max
respectively, another function is
defined to compute the relative membership degree,
which is called as the weak relative membership
degree shown in ⒅:
⎧
C ( X )
⎪rC
2
( X ) = 1-
⎪
C +C
max min
⒅
⎪
T ( X )
⎨ r ( X ) = 1-
T 2
⎪
T +T
max min
⎪
Q(
X )
⎪
r ( X ) = 1-
Q 2
⎩
Q +Q
max min
Finally, the relative membership degree based on
influence degree coefficients e ( ec
, et
, eq
) can be
obtained by linearly superposing ⒄ and ⒅, shown
as ⒆:
⎧ r ( X ) = e r ( X ) + (1 − e ) r ( X )
C
c C 1
c C 2
⎪
⎨ r ( X ) = e r
1
( X ) + (1 − e ) r
2
( X )
⒆
T
t T
t T
⎪
⎩r
( X ) = e r ( X ) + (1 − e ) r ( X )
Q
q Q 1
q Q 2
When e ( ec,
et
, eq
) is equal to e (1,1,1 ) , ⒆ is
becoming ⒄, which expresses the objective changing
range has the most great influence on the optimization
result.
From ⒄ , ⒅ and ⒆ , the objective eigenvalue
matrix described as ⒁ can be transformed into the
relative membership degree matrix described as ⒇:
⎡r
( X ) r ( X ) ... r ( X ) ⎤
C 1 C 2
C M
⎢
⎥
⒇
⎢r
( X ) r ( X ) ... r ( X )
T 1 T 2
T M ⎥
⎢
⎥
⎣r
( X ) r ( X ) ... r ( X )
Q 1 Q 2
Q M ⎦
3) Relative fitness function based on relative
membership degree
Extracting out the maximal value and the minimal
value from each row of the matrix described as ⒇,
which are marked as r and r b
:
g
M
M
M
r = { r , r , r } = {maxr
( X ),maxr
( X ),maxr
( X )} = {1,1,1} (21)
g
gC
gT
gQ
i=
1
C
i
i=
1
r = { r , r , r
M
M
M
} = {min r ( X ), min r ( X ), min r ( X )} = {0,0,0} (22)
b
bC
bT
bQ
i=
1
C
i
i=
1
T
T
i
i
i=
1
i=
1
Q
Q
i
i
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