Page 182 - Mathematical Models and Algorithms for Power System Optimization

Discrete Optimization for Reactive Power Planning 173

programming method are shown in Appendix A, and the method only changes one integer
each time.
(3) The improvement procedure is then used to enhance this solution further, but no
improvement solution is found in this iteration. Thus, the solution obtained is only by the
approximation method, and first iteration ends with an integer solution in the line with
method of P and cost as 20.8. The integer solution has not been improved after it becomes
feasible.

Second iteration:

(1) With the integer solution from the first iteration as the starting point, approximation
integer programming method will be used to solve MILP problem after the solution is
linearized. Its integer solution changes in both the second and third stage. The obtained
integer solution is shown in line 5 and cost as 13.3.
(2) As the integer solution improved algorithm that can change two integers each time is used,
the solution may be further improved. After several substeps of improvement, seven
installed nodes have been decreased to five installed nodes, that is, three installed adjacent
nodes 102, 106, and 107 are reduced to one installed node, 106. According to the result of
the iteration, the cost is 13.3 million yuan.

Third to seventh iteration:

With the integer solution of the second iteration as the starting point, the execution procedure is
the same as that of the second iteration. However, from the third to seventh iteration, even
continuous variable convergence and integer variables no longer change and improve, except
for the number of reactive capacitor compensations at node 106, which was reduced by one
bank in Step 3. This means that the integer solution obtained in the second iteration is a
nonlinear power flow solution meeting problem P.

The results of Cases 2–5 are briefly explained as follows. Cases 2 and 3 have the same voltage
magnitude bounds of 0.95–1.05 and the same capacitor candidate set. The difference between
Cases 2 and 3 is that Y i i 2 E C Þ is decreased and Y i i 2 N C Þ is increased in Case 3 as indicated by
ð
ð
mark superscript a in Table 6.2 in which both cases are successfully solved with almost the
same CPU time.
Cases 4 and 5 have the same conditions as Cases 2 and 3, respectively, except for voltage
magnitude bounds of 0.97–1.03. The change of voltage bounds makes more violations occur,
and more violations may cause larger computation time and more capacitor installations.
Compared with Case 2, Case 4 is successfully solved with almost the same computation time,
although more initial violations exist in Case 4 than in Case2. As for Case 5, the number of
existing capacitor units is decreased, and it takes longer computation time to adjust new
capacitor units.