Page 165 - Mathematical Models and Algorithms for Power System Optimization

156 Chapter 6

only provide a local optimal solution, that is, an optimal solution around the initial value. It is also
clearly indicated in the MINOS User Manual of Stanford University that the solution obtained is
the optimal solution around the initial value. In that way, the author failed to obtain a feasible
solution with MINOS when using straight start-up of power flow as the initial value. All optimal
solutions mentioned in this chapter refer to local optimal solutions. It is possible to determine the
initial values for different discrete solutions using different methods based on the continuous
solution. In this chapter, the influence of different initial values on the algorithm is discussed.

6.2.6 Verification of the Correctness of Discrete Solutions

The order of magnitude of nodes number Nin power system is normally 102–103. The number
of constraints for optimization calculation is approximately 2N.

Most of the constraints are nonlinear constraints, and the number of variables is about 3N,in
which about one-third account for integer variables. At present, it is difficult for a precision-
integer programming optimization algorithm to deal with problems of such large scale. IBM’s
MILP package can only solve problems with fewer than 50 integer variables. Common
nonlinear algorithms cannot deal with more than 200 nonlinear constraints either.
To validate the procedure developed and the discrete solution obtained, several comparisons
are made in this chapter as follows:

(1) Verification of differential value calculation formula: To validate the procedure developed
with the algorithm proposed in this chapter, all differential value calculation formulas for the
calculation procedure are verified with the original differential value calculation formula.
(2) Verification of the discrete optimal solution: There are not many new algorithms for
discrete optimization. Moreover, a precision discrete optimization algorithm cannot be
used to deal the real-scale discrete optimizations. Thus, it is necessary to validate the
discrete optimization solution obtained with the new algorithm. Generally, for the same
problem, the objective function value of a continuous optimization solution obtained by
the same procedure should be the smallest, the objective function value of discrete
optimization solution should be in the middle, and the objective function value of the
truncated solution should be the largest. The results in this chapter support these criteria.
(3) Comparison with results from MILP algorithm programming software package: If the
number of integer variables is less than 50, the linear integer programming with
calculation procedure based on the algorithm proposed in this chapter can reach the same
solution as that of an MILP algorithm programming software package.
(4) Comparison with results from MINOS nonlinear software package: Optimal power flow
(OPF) discrete solution obtained with the algorithm proposed in this chapter will be taken