Page 39 - Decision Making Applications in Modern Power Systems

16 Decision Making Applications in Modern Power Systems

The concordance and discordance matrices are expressed in terms of
the concordance and discordance indices. The concordance matrix C
and discordance matrix D are given by
2 c 12 ? c 1m
2 3
6 c 21 2 ... c 2m 7
6 7
C 5 6 ^ ^ ^ ^ 7 ð1:21Þ
6 7
^ ^ ^ ^
4 5
? 2
c m1 c m2
2 3
2 d 12 ? d 1m
2
d 21 ... d 2m
6 7
6 7
D 5 6 ^ ^ ^ ^ 7 ð1:22Þ
^ ^ ^ ^
6 7
4 5
? 2
d m1 d m2
It is to be noted that the value elements of D are undefined for k 5 l,
and it is nonsymmetric square matrix.
Step 5: Calculating the concordance and discordance dominance matrices
This step involves the use of a threshold value. The threshold value is
a critical value that is involved in the formation of dominance matrices.
^
Let ^ c and d be the threshold values that are calculated as an average
value of the concordance and discordance indexes.
m
1 X X m
^ c 5 l51 c kl ð1:23Þ
mm 2 1Þ l 6¼ k
ð
k51
k 6¼ l
m m
1
^
X X
d 5 l51 c kl ð1:24Þ
mm 2 1Þ l 6¼ k
ð
k51
k 6¼ l
The concordance and discordance dominance matrices are calculated
based on the comparison of concordance and discordance indexes and
^
threshold values (^ c and d). Let cd kl and dd kl be the elements of the matri-
ces. Mathematically, it is given by

1if c kl $ ^ c
cd kl 5 ð1:25Þ
0if c kl # ^ c
and
^ ^
(
dd kl 5 1if d kl $ d ð1:26Þ
0if d kl # d ^