Page 174 - Mathematical Models and Algorithms for Power System Optimization

Discrete Optimization for Reactive Power Planning 165

6.3.3 Single-State Algorithm for Discrete Optimal VAR Optimization

6.3.3.1 The main computational procedure
ThewaytosolveproblemPistolinearizeproblemPintheiterationproceduretocreatethelinearized
problem PL. Meanwhile, theapproximation method ofmixed-integer linear programming isused to
solve the linear problem. The following section discusses the calculation steps:

Step 1: Formulate the VAR planning problem as a nonlinear MIP problem (or MINLP
problem) described as problem P in Section 6.3.2.2.
0
Step 2: Set k¼0, and assume an initial value Z based on power flow solution of the system,
and vector Z defined as:

ð
Z ¼ X, Y, WÞ ¼ U, θ, P G , Q G , T, C, Wð Þ (6.14)
0
This initial value z can be obtained by relaxing the limits for V, θ and T and setting Y, W to
zero when repeatedly solving linearized LF equations by LP. Certainly, the initial value can
also be obtained by any load flow solution method, but only the sequence of variables
should be V, θ, T, Y.
Step 3: Construct a linear MIP problem (or MILP problem) PL by linearizing load flow
k
Eqs. (6.5)–(6.8) at point z , where PL is the problem with the same objective function and
constraints as problem P, except that Eqs. (6.5) and (6.6) are linearized.
Step 4: Obtain an integer-feasible solution of the linear MIP problem PL and improve the
obtained solution by using the approximation method in Appendix A. In searching for an
integer-feasible solution, this method treats integer variables as fixed values, that is, as
nonbasic variables, then only one integer variable is increased by +1 or decreased by 1
each time when repeatedly using the simplex method for LP.
Step 5: Obtain a better integer solution than that of Step 4 by changing the values of two
integer variables each time to decrease the installation cost further and to install capacitors
at fewer nodes. The detailed steps for the improved algorithm in Step 5 are shown in
Section 6.3.3.2.
k
Step 6: Test for convergence if Z 6¼Z k+1 , make k k+1, then go to Step 3; otherwise, go to
next step.
k
Step 7: Terminate the algorithm because z is an optimal solution of problem P.

In fact, this algorithm can be effectively executed and can quickly find an integer-feasible
solution by the approximation method in Step 4 even if Step 5 is not executed. However, based
on existing experience, the solution from Step 4 was somewhat an impractical solution, that is,
it might allocate only one capacitor unit to each of several adjacent nodes. This makes the total
cost increase due to fixed costs.