Page 219 - Mathematical Models and Algorithms for Power System Optimization

210 Chapter 6

X 2 X 2
F ¼ f + λ ð voltage deviationÞ + λ ð reactive power deviationÞ (6.49)

6.6.4 GA-based Algorithm for Discrete VAR Optimization

As mentioned earlier, GA is a random search algorithm with the ability to obtain the global
optimal solution, thus it may be applied to the discrete VAR optimization problem. However, if
the GA search process is applied directly without considering the relations among VAR
sources, transformers, and voltages in power system, there will be a lot of blind points, and the
search efficiency will be very low. This section attempts to combine the expert rules and GA
technology to improve the efficiency of GAs. The main points are as follows.
6.6.4.1 String performance of integer variables

(1) Integer variables are represented by the decimal system, whereas GA handles variables
based on the binary system. Thus, in this section, if GA is based upon the decimal system,
the solution space in the practical system may be reduced. This section employs the
following method to handle integer variables:

(
if Y i ¼ 0, then Y i ¼ Y i

Y i ¼ Y i λ (6.50)
if Y i 6¼ 0, then Y i ¼ Y i
where Y i —upper limit of each integer variable; Y —lower limit of each integer variable;
i
λ—random numbers evenly distributed in [0,1]; Y i —integer by rounded-off Y i λ; if the
rounded-off Y i ¼0, then let Y i equal to the lower limit.
For example, if the upper and lower limits of tap ratio is [1,17], then Y i ¼ 17, and if the random
number λ ¼0.789, the random tap position of transformer is set as:

Y ¼ Y λ ¼ 17 0:789Š ¼ 13:413Š ¼ 13
½
½
Considering the fact that multiple-integer-feasible solutions in a system may not be of great
difference. For instance, the initial feasible tap ratio of the earlier discussed transformer is 1,
and the final feasible tap ratio solution might be 17, and thus, after obtaining the initial feasible
solution, it is possible to narrow down the random variation into:

0 0
Y + N,if Y + N > Y
∗
Y ¼ 0 (6.51)
Y,if Y + N < Y
0 0
∗ Y + N,if Y + N < Y
Y ¼ 0 (6.52)
Y,if Y + N > Y
∗
∗
where N ranges from 1 to 3, Y is the new lower limit, and Y is the new upper limit. After
changing the new limits, the random integer variables are also changed using Eq. (6.50).