Page 308 - Mathematical Models and Algorithms for Power System Optimization

300 Chapter 8

Eqs. (8.14)–(8.17) are only valid for active bus, and u i ¼0 when there is no control action, so
that the decoupled reference {δ ei , U ei , α ei } could be solved. If Eqs. (8.14)–(8.16) derive
differentiation against the time, then the following equations could be derived:

∂G ei ∂G ei _ ∂G ei _ ∂G ei
¼ δ ei + U ei + _ α ei ¼ 0 (8.18)
∂t ∂δ ei ∂U ei ∂α ei
∂G ePi ∂G ePi _ ∂G ePi _ ∂G ePi
¼ δ ei + U ei + _ α ei ¼ 0 (8.19)
∂t ∂δ ei ∂U ei ∂α ei

∂G eQi ∂G eQi _ ∂G eQi _ ∂G eQi
¼ δ ei + U ei + _ α ei ¼ 0 (8.20)
∂t ∂δ ei ∂U ei ∂α ei
_ €
δ ei , U ei , _ α ei could be solved from Eqs. (8.18)–(8.20).
Based on the definition for the observation decoupled state space and the existing observation
decoupled reference, the following equations can be written:
_ _ _
δ i ¼ δ i δ ei , δ i ¼ δ i δ ei , U i ¼ U i U ei , α i ¼ α i α ei (8.21)
n _ o
Then the observation decoupled state coordinate δ i , δ i could be solved, and Eqs. (8.14)–(8.17)
could be rewritten into the following forms by means of Eq. (8.21):

0
i
E
ð
½
G i ¼ P mi U i U i sin δ i δ i α i α i Þ u i ¼ 0 ð i 2 1, NŠÞ (8.22)
X 0
di
0
i
E
G Pi ¼ U i U i sin δ i δ i α i α i Þ
ð
X 0
di
N
U j °
X
C ij U i U i sin α i α i α j + φ 90
ij
Z ij
j ¼ 1
(8.23)
j 6¼ 1
N cosφ
X
2 ij
½
ð
U i U i C ij P Li ¼ 0 i 2 1, NŠÞ
Z ij
j ¼ 1
j 6¼ i
2
0
i
E U i U i
G Qi ¼ U i U i cos δ i δ i α i α i Þ
ð
X 0 X 0
di di
N
X U j °
+ U i U i cos α i α i α j + φ 90
C ij
ij
Z ij
j ¼ 1
(8.24)
j 6¼ 1
N sinφ
2
X ij 1
U i U i C ij Q Li ¼ 0 i 2 1, N½ð ŠÞ
Z ij X cij
j ¼ 1
j 6¼ 1