Page 312 - Mathematical Models and Algorithms for Power System Optimization

304 Chapter 8

The first derivative of time t for V is required to be less than zero:

i
X X h _
_
€
V¼ V i ¼ δ i δ i + δ 50Þ δ 50ð Þ 0 (8.40)
ð
i i

The sufficient condition for V monotonous decreasing is that each component V i of V should be
less than 0, that is:
h i
_ _ €
V i ¼ δ i δ i + δ 50ðÞ δ 50ð Þ 0 (8.41)
i i
Substitute Eq. (8.38) into Eq. (8.41); the forms of the norm reduction control criterion in the

space δ, δ 50ð Þ could be derived as follows after sorting and moving, etc.:
_ 0
8
> J i δ i δ i E U i
> i _ _
ð
ð
> u i + P mi sin δ i α i Þ P Di δ 50ð Þ δ 50Þ > 0
> _ 0 i i
< X
δ 50ðÞ
i di (8.42)
_
> 0
i
> J i δ i δ i E U i
> _ _
ð
ð
> u i + P mi sin δ i α i Þ P Di δ 50ð Þ δ 50Þ > 0
: _ 0 i i
δ 50ðÞ X di
i

(2) The form of the norm reduction control criterion in the space δ, δ s . Let
_
W 1i ¼ δ ¼ δ i δ ei , W 2i ¼ δ si ¼ ω i ω s , U i ¼ U i U ei , α i ¼ α i α ei (8.43)
where
X
J j ω j
ω s ¼ X
J j

Then the dynamic equation in the space δ, δ s is of the following form:
8
_ _ _
> W 1i ¼ δ i ¼ W 2i δ ei ω 0 + ω s
>
>

< 1 0
_ € E
i
ð
½
W 2i ¼ δ si ¼ P mi U i + U ei sin W Li + δ ei α i + α ei ފ P Di W 2i ω 0 + ω s Þ
ð
J i
> X 0
> di
>
:
_ ω s ð i 2 1, NŠÞ
½
(8.44)
Define the norm as:
X
X 1 2 2
V ¼ V i ¼ δ + δ _ si (8.45)
i
2
The first derivative of time t for V is required to be less than zero:
X
X _
_ €
V¼ V¼ δ i δ i + δ si δ si 0 (8.46)