Page 269 - Mathematical Models and Algorithms for Power System Optimization

Optimization Method for Load Frequency Feed Forward Control 261

To transform the differential equation into difference equation, it is equal to require a retainer
set after the sampling switch, and the following direct Z transform is utilized:

TS
1 e
GZðÞ ¼ ZG SðÞ (7.108)
S
where

PSðÞ
GSðÞ ¼
QSðÞ
When the characteristic equation Q(S)¼0, it does not include the n-order singularity, the
following equation or Z transform can be used:

ð
X PS K Þ 1 Z T
GZðÞ ¼ (7.109)
Z
0
Q S K Þ 1 e S K T 1
ð
1
where S K is the root of characteristic equation Q(S), Q 1 (S)¼Q(S) S, and T is the sampling time.
When the characteristic equation Q 1 (S)¼0 and it has the n-order singularity, for instance, the
1 1 PSðÞ
transfer function including the factor or n , can be expanded into partial fraction
S n ð S + aÞ aSðÞ
equations by the partial fraction expansion method

After G(S) has been expanded into partial fraction equation, the single root part can be
transformed directly into Z transform based upon Eq. (7.109). As for the transfer function
1
including n factor can be transformed using the following equation, that is, when
ð S + aÞ

1
0
G SðÞ ¼
n +1
ð S + aÞ
the Z transform is

n
0 n 1 ∂ 1 TS
G ZðÞ ¼ Z 1ð Þ 1 e
n! ∂a n S + a
n
1 ∂ 1
n TS
ð
¼ 1Þ Z 1 e (7.110)
n! ∂a n S + a
1 ∂ n 1
n T
¼ 1Þ 1 Z
ð
n!∂a n 1 e aT 1
Z