Page 94 - Mathematical Models and Algorithms for Power System Optimization

84 Chapter 4

Somereferencessuggestpowerflowcalculationsshouldbeformulatedasnonlinearprogramming
(NLP) problems. The direction and step size in the power flow calculation are determined by
minimizing the objective function. However, under ill conditions, this may still result in a local
solution, rather than a final convergent one due to the nonlinearity of the power flow problem.

This chapter explores the application of the SA technique in solving the ill-conditioned power
flow of a power system and proposes a combined algorithm based on both N-R and SA
methods. The SA technique is used to probabilistically determine the corrected step size of
variables in the N-R method. The possible direction of a true solution is sought through random
searching to improve the convergence in ill-conditioned power flow calculations. The
numerical case study for the actual system shows that the algorithm proposed in this chapter is
an alternative algorithm for power flow calculation, which can indeed effectively improve the
convergence characteristics of large-scale ill-conditioned power flow.

4.1.4 Overview of Constrained Power Flow with Objective Function
(based on OPF Method)

With the growing complexity of the power system operation, the OPF problem becomes
increasingly important. For the OPF, the objective function generally is to minimize the
operating cost of the generator, and the constraint function generally is to satisfy the operating
condition of the power system. In numerous papers published in recent years, OPF is usually
treated as a nonlinear problem in which the transformer tap ratio, number of capacitor banks,
and number of reactor banks are all treated as continuous variables.

However, as some references point out, the transformer tap ratio, number of the capacitor
banks, and number of the reactor banks in OPF should be all treated as discrete variables.
Otherwise, the solution obtained by using these algorithms may not be the optimal or even
feasible after truncation. The OPF problem in this chapter is the same as the conventional one. It
is treated as a mixed-integer nonlinear problem in which the transformation ratio, number of
capacitor banks, and number of reactor banks are all treated as discrete variables. Therefore, the
OPF in this chapter can also be named as a discrete OPF.

The existing generalized benders decomposition (GBD) method can be used to solve the
mixed-integer nonlinear problem. However, the GBD method needs a large amount of
computational time to solve the large-scale discrete OPF problem, which is not practically
applied. Some mathematical programming methods also have certain difficulties in accurately
solving a discrete-sized OPF problem of actual scale.

To solve the discrete power flow problem of actual scale, this chapter uses the method and
concept of successive linear programming (SLP) to solve the mixed-integer LP problem in each
iteration process. As the discrete OPF is nonconvex, the optimal solution obtained using the
SLP technique is relative to the initial value.