Page 176 - Mathematical Models and Algorithms for Power System Optimization

Discrete Optimization for Reactive Power Planning 167

nodes are installed with reactive power compensation equipment, but each node has a small
compensation, is likely to occur. The reason is that only the most promising integer
variable can be processed once in the calculation procedure of the original algorithm,
adding or deducting the integer by 1, then solving LP to make the problem feasible. Such a
calculation procedure may lead to a small amount of reactive power investment on
each adjacent node. The need for increasing the fixed cost without connection to the
equipment capacity at the location of new installed reactive equipment leads to an
increment of total cost.

An improved algorithm is proposed in this section by changing two integers at the same time to
overcome the deficiencies of the original algorithm.

One of the integers is added by 1 or 2, whereas another integer is deducted, until the
latter becomes zero. In other words, when the integer-feasible solution has been solved
with the original algorithm, two nonzero integer variables will be comprehensively
considered and selected. If the problem is still feasible based upon the procedure previously
mentioned, and the objective function value decreases, this pair of integer variables will be used
to replace the original pair of integer solutions. This will continue until a new integer pair
cannot be obtained. From the perspective of system operation, it is workable to add reactive
power compensation equipment to one of two adjacent nodes, yet reduce reactive power
compensation equipment from another node, and this won’t lead to problem infeasibility.
However, reduction of a new reactive power node will help cut the fixed cost, so as to avoid
installing a small number of reactive power compensation equipment at adjacent nodes at the
same time.

The principle of the improved integer solution algorithm is to reduce the number of nodes
for the newly installed reactive power compensation equipment, replacing a pair of
nonzero integers with one nonzero integer and one zero integer. The detailed steps are
given as follows:

Substep 1: Express the feasible solution obtained in Step 4 as:
∗ ∗ ∗ ∗
Z ¼ X , Y , Wð Þ (6.15)
Substep 2: Stochastically select a pair of nonzero integer variables y u ∗ and y s ∗ from node u
and s, and adjust the number of capacitor banks installed at the two nodes, then recalculate
the new investment cost; if the cost is reduced, they will be taken as a new pair of
changeable integers, then go to the next step. Otherwise, it means that present integer
solutions cannot be further improved, and the calculation terminates.
Substep 3: Make the new integer variable vector Y n by increasing one of the integers,
y u ∗, by 1 or 2, and decreasing another integer y s ∗ by 1, then readjust the fixed cost vector W*
to satisfy Eq. (6.14). For example, if y s ∗¼1, then let y s ∗¼0, w s ∗¼0. Thus, the fixed cost at
node s becomes zero.