Page 70 - Mathematical Models and Algorithms for Power System Optimization

60 Chapter 3

μ xðÞ ¼ 0, f i xðÞ f i 0
fi
μ xðÞ ¼ 1, f i xðÞ f i 1
fi
0 1
In Eq. (3.11), f i is at an unacceptable level, whereas f i is at a fully desirable level; the
fuzzy membership function μ xðÞ reflects the membership degree of function f(x) between
fi
0 1
f i and f i .

3.4.2 Formation of Fuzzy Objective Index of GMS

The fuzzy performance index corresponds to the fuzzy goal. Depending on the OPindex (0,1)
interval of system operation index, the GMS may consider two fuzzy performance indexes, that
is, reserve margin and production expenses. When the system is at a high load level, fuzzy
performance index with reserve margin may be selected as objective index. When the system is
at a low load level, either the fuzzy performance index with reserve margin or with production
expenses may be selected as objective index.

(1) Fuzzy membership function with reserve margin as objective index is expressed as:

8
0, P RM i, tð Þ < R mn
>
>
<
R m iðÞ R RM i, tÞ
ð
ð
μ i, tÞ ¼ 1 W rm , P RM i, tð Þ R ma (3.12)
gl
R m iðÞ R iðÞ
> m
>
:
1, P RM i, tÞ R ms
ð
where P RM (i,t)—minimum surplus capacity when reaching to the state (i,t), W rm —weight
coefficient of reserve margin, R ma —acceptable minimum reserve capacity, R ms —reserve
capacity to the reliability requirements, R m iðÞ max P RM i, tð Þg;
f
T
R iðÞ min P RM i, tÞg:
f
ð
m
T
(2) Fuzzy membership function with production expenses as objective index is expressed as:
ð
C P i, tÞ C iðÞ
P
μ ð i, tÞ ¼ 1 W cp (3.13)
g2
C P iðÞ C iðÞ
P
where C P (i,t)—minimum production expenses when reaching to the state (i,t),
f
W cp —weight coefficient of production expenses, C P iðÞ max C P i, tð Þg;
T
C iðÞ min C P i, tÞg:
ð
f
P
T
The fuzzy membership function curve of production expenses is shown in Fig. 3.3.