Page 122 - Mathematical Models and Algorithms for Power System Optimization

112 Chapter 4

(2) The detailed coefficients of the 5-bus system example in the mathematical model: In the
following are the expressions for the 5-bus system examples corresponding to the
continuous variable vector X, integer variable vector Y, continuous variable coefficient
matrix A, discrete variable coefficient matrix B, and right hand side vector b in Eq. (4.32):

ð
X ¼ U 1 , θ 1 , U 2 , θ 2 , U 3 , θ 3 , U 4 , θ 4 , U 5 , θ 5 , P GI , Q G1 , Q G2 Þ
ð
Y ¼ Y T1 , Y T2 , Y C1 , Y C2 , Y C3 Þ
2 3
J 11 J 14 A P1 A Q1
6 7
J 22 J 25
6 A Q2 7
6 7
A ¼ 6 J 33 J 34 J 35 7
6 7
6 7
J 41 J 43 J 44 J 45
4 5
J 52 J 53 J 54 J 55
2 3
A T1
6 7
A T2
6 7
6 7
B5 6 A C1 7
6 7
6 7
A T4 A C2
4 5
A T5 A C3
T
½
b ¼ ΔP 1 , ΔQ 1 , …, ΔP 5 , ΔQ 5 Š
where

∂P i =∂U i ∂P i =∂θ i ∂P i =∂U j ∂P i =∂θ j
J ii ¼ , J ij ¼
∂Q i =∂U i ∂Q i =∂θ i ∂Q i =∂U j ∂Q i =∂θ j

∂p i =∂p i 1 0 0
A pi ¼ ¼ , A Qi ¼ ¼
0 0 ∂Q i =∂Q i 1

∂P ij =∂T i ∂P ji =∂T i 2
A Tij ¼ , A Tji ¼
, A Ci ¼ U q Ci
i
∂Q ij =∂T i ∂Q ji =∂T i
If U 0 ¼1.0 and θ 0 ¼0, the coefficients of A, B, and b are:
2 3
66:67 0 0 0 0 0 66:67 0 0 0 1 0 0
0 66:67 0 0 0 0 0 66:67 0 0
6 0 1 07
0 0 33:33 0 0 0 0 0 33:33 0
6 7
6 0 0 0 7
0 0 0 33:33 0 0 0 0 0 33:33 0 0 1
6 7
6 7
6 0 0 0 0 22:8904 4:8247 3:9794 0:2865 18:911 4:539 0 0 0 7
A5 6 7
0 0 0 0 4:8255 22:8898 0:2865 3:9794 4:539 18:911 0 0 0
6 7
6 7
66:67 0 0 0 3:9794 0:2865 85:9404
6 4:8737 15:291 4:58720 007
6 7
0 66:67 0 0 0:2865 3:9794 4:8737 85:9328
6 4:5872 15:291 0 0 07
0 0 33:33 0 18:911 4:539 15:291 4:5872 67:532 9:1252 0 0 0
4 5
0 0 0 33:33 4:539 18:911 4:5872 15:291 9:1262 67:5372 0 0 0