Page 167 - Mathematical Models and Algorithms for Power System Optimization

158 Chapter 6

6.2.8 Way of Processing Objective Function

If capacitor investment minimization is taken as the objective function, the number of variables
of T (transformer tap) and C (capacitor capacity and bank number) may be increased, and the
coefficient G ij and B ij of power flow equation may be modified so as to turn the equations of G ij
and B ij into the equations of T and C, obtaining the partial derivatives of equations T and C, and
X
to calculate out the investment on capacitor C: min ð d i W i + c i C i Þ.
i2N C
6.2.9 Way of Processing Transformer Tap T and Capacitor Bank C

The power flow equation of transformer line and capacitor line should be modified, so as to
obtain the partial derivatives for T and C, and to establish the linear equations containing T and
C. The detailed modification method is shown in Appendix B.

6.3 Single-State Discrete VAR Optimization

6.3.1 Outline

As for single-state discrete reactive power optimization, its objective is to determine the
minimum cost expansion plan of new reactive sources so as to guarantee feasible operations
under a single-state operating mode. The basic feature is that, when solving large-scale reactive
power compensation equipment planning problems, the number of capacitors may be taken as a
discrete variable.

As power system problems have better linearity around a static stable point, the MILP method
may hereby be used to solve its linearized integer programming problem and its nonlinear
problem through iteratively executing LP.

The algorithm here employs an excellent approximation method for solving linear MIP
problems because an exact optimal solution of MIP problems can hardly be obtained.
Furthermore, to solve a nonlinear MIP problem, the algorithm exploits the concept of recursive
LP so that it is capable of obtaining an AC load flow solution of the VAR planning problem.
The program can easily be developed based on this algorithm by incorporating the options of
existing LP software.

It should be pointed out that an integer-feasible solution can be quickly obtained by the
approximation method, but it is a somewhat impractical solution because there is only one
capacitor unit to be installed in each of several adjacent nodes. Hence, an improved procedure is
proposed to remedy this impractical solution.

Thissectiontreatsonlynormaloperationswithoutconsideringcontingencystates.Thealgorithm
developed here provides a fundamental mathematical tool for multiple states. The extension of
the algorithm from a single-state problem to a multistate problem is described in Section 6.4.