Page 16 - Mathematical Models and Algorithms for Power System Optimization

6 Chapter 1

(2) Considerations for the use of the existing solution tools
A linear programming-based algorithm can be applied because many excellent software
packages are available. To solve large-scale optimization problems, it is better to take
advantage of the existing calculation conditions, because the computing time is related to
the dimension of the models. From a mathematical perspective, a linear programming
method is normally suited for the problems of large-scale power systems.
A large-scale discrete optimization algorithm is difficult to solve even with today’s rapid
advancement of computer performance. From a point of pure mathematics, the states of all
of the equipment in a power system—with its hundreds to tens of thousands of nodes—
needs to be represented with integer variables, which makes the calculation time to be
difficult to bear. Therefore, it becomes necessary to develop approximation integer
programming algorithms based on the existing software packages, so that discrete
optimization calculations for a large-scale power system becomes possible.
A nonlinear optimization method also requires a practical approximation algorithm. Some
nonlinear models of power system optimization are derived by a linearization method, of
which some variable coefficients are derived from the first and second derivative of the
transcendental functions of the power flow equation. The precondition for a derivative-
based optimization algorithm is that these transcendental functions are continuously
differentiable, which is a very hard condition even if difference function is proved to be
able to approximately represent differential function. In practice, it’s been proven that
some approximation processing methods based on the existing software packages can
also satisfy the requirements of engineering precision.

(3) Considerations for the local solution
Any optimization algorithm has a problem of the local solution. Optimization algorithms
can be divided into two categories, namely, numerical and non-numerical method.
However, both methods have a problem with local solutions, that is, the optimal point can
only be obtained around the current starting point. For example, the method for calculating
the discrete reactive power optimization model is based on a successive linearization
method.However,consideringthediscretevariable,themodelcannotguaranteeconvexity.
As for nonconvex optimization, different initial values may give different solutions.
An expert system approach could help screen and filter some obviously unreasonable
optimization search directions. Some actual problems cannot be solved by analytical
mathematics but may be solved with an expert system approach. For instance, in discrete
reactive power optimization, the expert system approach could help determine the
direction of continuous variable truncation, so as to derive the integer feasible solution
more quickly. A combination of the stochastic searching method, expert system approach,
and traditional analytical algorithm could significantly reduce calculation time and obtain
a reasonable discrete solution, efficiently processing the integer variables for some
practical optimization problems.