Page 99 - Mathematical Models and Algorithms for Power System Optimization

New Algorithms Related to Power Flow 89

4.3.2 Formulation of Power Flow with Quadratic Function

Bus power equations for the power flow calculation in form of polar coordinates are as follows:
8 X

> P i ¼ U i U i G ij cosθ ij + B ij sinθ ij
>
>
j2i
<
X

>
> Q i ¼ U i U i G ij sinθ ij B ij cosθ ij
>
:
j2i
Modified into nonlinear equation sets:
8 X

> ΔP i ¼ P is U i U i G ij cosθ ij + B ij sinθ ij ¼ 0
>
>
< j2i
X

>
> ΔQ i ¼ Q is U i U i G ij sinθ ij B ij cosθ ij ¼ 0
>
:
j2i
To apply the SA technique to solve power flow problems of power systems, the modified bus
power equations of power flow calculation are used to construct a quadratic function that is
similar to the nonlinear objective function. The detailed mathematical model with quadratic
objective function is as follows:
n o
X 2 X 2
min FXðÞ ¼ ΔP XðÞ + ΔQ XðÞ (4.1)
i i
ΔP i XðÞ ¼ P i XðÞ P is , i 2 N (4.2)
ΔQ i XðÞ ¼ Q i XðÞ Q is , i 2 N (4.3)
X
P i XðÞ ¼ P ij XðÞ (4.4)
X
Q i XðÞ ¼ Q ij XðÞ (4.5)
P is ¼ P iG P iL
(4.6)
Q is ¼ Q iG Q iL
2
P iL ¼ P iLo a P U + b P U + c P Þ
ð
2

Q iL ¼ Q iLo a Q U + b Q U + c Q (4.7)
a P + b P + c P ¼ 1:0
a Q + b Q + c Q ¼ 1:0

4.4 Calculation Procedure of SA based N-R Method

Based on the original calculation procedure of the N-R method, it is easy to implement the new
program for the new combined algorithm. The following steps outline the detailed solution
procedures: