Page 101 - Mathematical Models and Algorithms for Power System Optimization
P. 101
New Algorithms Related to Power Flow 91
where J is the same Jacobian matrix as in the N-R method and M(X k ) is an unbalanced
power vector.
Step 2: Select a random number r:
½
r ¼ random 0, 1
Step 3: Generate a new ΔX:
ΔX ¼ rΔX k
Step 4: Move the current system structure status to a new one:
X k +1 ¼ X k + ΔX
Step 5: Judge objective function values F(X k +1) and F(X k ). If ΔF¼F(X k+1 ) F(X k )
<0 or exp( ΔF/T)>random[0, 1], then accept new X k+1 ; otherwise reject this X k+1 ,
return to Step 2 to select a new random number r, then recalculate X k+1 .
n o
X 2 X 2
min FXðÞ ¼ ΔP + ΔQ
i j
(4) Cooling scheme:
1. Set the initial temperature. To determine initial temperature T 0 , first calculate the
+
balanced increment ΔF of the objective function after a set of random movements,
then obtain T 0 by solving the equation:
a 0 ¼ exp ΔF+ =T 0 Þ (4.8)
ð
where a 0 is an acceptable value equivalent to the ratio of feasible status transfer
number to proposed movement number in each temperature range. At the beginning of
iteration, typically take a 0 ¼0.75–0.85; in other words, at this point, at least 75%–
85% of transfers must be accepted. From Eq. (4.1), T 0 can be derived from:
+ 1
T 0 ¼ ΔF = ln a 0 (4.9)
2. Set the cooling rate. Typically, cooling rate b(T k ) is as follows:
bT k ¼ 0:85 0:95
ðÞ
ðÞT k
T k +1 ¼ bT k
3. Final stopping criterion:
T k < T min orFX k +1 Þ FX k < ε orMX k < ε 0
ðÞ
ðÞ
ð
4. Maximum number of movements in each temperature range: Typically, the maximum
number of movements N max can be determined according to actual problems. For an
ill-conditioned system, N max is a very loose number. When the sum of the numbers of
