Page 217 - Mathematical Models and Algorithms for Power System Optimization

208 Chapter 6

6.6.2 Necessity of Applying Artificial Intelligence Algorithms

The optimization methods can be divided into two broad categories, that is, an analytic method
and a direct method. The former is to employ the first- and second-order derivatives of
functions in the solution of n-dimension extremum, and the latter is not to use such derivatives.
Because the derivative of the function indicates a change in rules of function values, it is natural
to utilize such derivatives of the function when solving the extremum value. The specific
approach to the analytic method is to solve a set of nonlinear equations with the derivative of
objective function as 0.

However, the analytical expression of objective function in practical problems is rather
complicated; it is difficult to find a derivative for some, and some only give the corresponding
relation between variables and objective functions. Hence the direct method must be adopted
to solve the extremum value of such functions, searching for the optimization solution of
function values in the possible descent direction in a maximum gradient. In fact, some direct
methods will take advantage of the analytic properties of the function, whereas the analytic
method can also be implemented only with the function values. Therefore, the two methods
cannot be separated completely.

Regardless of their effectiveness in many cases, the two methods are local optimal ones, or
rather the optimal points obtained are only the local ones around the existing points. For the
nonconvex optimization problem, different initial values may give different solutions, and
further improvement of objective functions must be able to resume stochastically or adopt other
means to enable the search process to proceed around another peak value. The prerequisite of a
derivative-based analytic method is that the function shall be continuously differentiable,
which is a very harsh condition, but still a strict one even allowed to use the difference
approximation.
In this section, the coefficients of tap ratio and capacitor bank number are approximately
represented by the differential values. Actually, the parameter space is not smooth, and thus, the
locally search optimization method is of a certain approximation.

The enumeration method is quite straightforward: in a finite finding space or a discrete finite
space, the search algorithm seeks from the objective function of each point at a time. Because
the power system optimization is a typical large-scale optimization, this algorithm cannot be
appliedas itsefficiencyis toolow.The dimensionalityofdynamicprogrammingalgorithmistoo
large to be solvable for the power system problems. Those to be handled in the power system are
generally large-scale problems, which is difficult to be solved using the existing algorithms.
Therefore, it is generally acceptable to find a near-optimal solution with practical values.

To sum up, when fully aware of the shortcomings of the differential method and enumeration
method, many researchers are trying to solve the local optimum problem in the optimization