Page 96 - Mathematical Models and Algorithms for Power System Optimization

86 Chapter 4

Sometimes, the practical system will still operate (i.e., a power flow solution exists), but the
existing power flow algorithm is likely unable to lead to a convergent solution, which is called
an ill-conditioned power flow.

A power system with such features may be described by a set of ill-conditioned nonlinear
algebraic equations. In other words, a small change in parameters will produce a large change in
the solution. When power systems are ill conditions, the existing algorithms can hardly ensure
convergence of the power flow calculation. Ill-conditioned power problems have been
investigated in many publications. However, most of these methods were either designed for
a specific ill condition or required substantial modifications from the standard load
flow method.

A new load flow method for solving ill-conditioned systems is proposed. The goal of
this proposed method is two-fold: (1) this method should be easily incorporated into the
existing load flow method, and (2) this method should be able to address different types of
ill conditions. To achieve this goal, the proposed load flow technique is a hybrid model
between the N-R and SA methods. The SA technique is employed to determine proper
correction steps of N-R solution variables, such that these variables will not be trapped into
a local solution.

One of the important advantages of the SA method is that it can selectively accept “uphill
movement”, that is, the objective function moves in an increasing direction to avoid falling into
a local solution. On this basis, this chapter combines the SA and N-R methods to formulate a
new combined algorithm to solve ill-conditioned power flow problems. The new algorithm
determines the appropriate step size of variables in the N-R method using the SA technique to
prevent them from falling into a local solution, so as to improve the convergence problem of the
ill-conditioned power flow calculation and minimize divergence occurrence. For the power
flow calculation of a normal system, the algorithm proposed in this section has the same
convergence characteristics as the N-R algorithm. For ill-conditioned systems where the N-R
method is difficult to achieve convergence, the new algorithm may lead to a convergent
solution as long as the solution exists.

As commonly known, any probabilistic method for solving stochastic processes is effective for
samples of a certain scale, the effect of which is not obvious for small samples. The algorithm
proposed in this chapter is no exception. In processing small cases with an infinitely large single
unit or just a few buses, this algorithm and the N-R method have the same critical convergence
point with increasingly heavy operation conditions of the system. Therefore, in terms of the
effect of theoretical and actual calculation, the algorithm proposed in this chapter is effective
for systems of a certain scale in processing ill-conditioned power flow problems. Because
typical real systems in a certain scale are sufficient to meet the requirement of this algorithm,
these systems are solvable with this algorithm.