Page 12 - Mathematical Models and Algorithms for Power System Optimization

2 Chapter 1

numerical or non-numerical procedures. To integrate more new elements and meet new
development trends, new models should be developed using existing and newly developed
methods based on the current computer technology to satisfy the actual requirements of the
power system, which require a solid mathematical foundation and engineering background
knowledge. The development of the power system optimization model is full of dialectic
wisdom. The authors’ main ideas are as follows:
(1) The optimization modeling is to transfer a practical problem into a mathematical problem
and obtain a feasible solution for the practical problem considering the existing
conditions. It has to fully consider the compromise among conflicting goals, such as the
simple model versus the complex calculation procedure, and vice verse, nonlinear model
versus linearized solution, and large-scale discrete optimization model versus the current
computing condition.
(2) Whether the developed model is solvable must consider many problems from the
theoretical perspective. For example, the problem with local solutions for any
optimization algorithm, which can be avoided by the method of the multi-point search
in the solution space. As for the nonconvex problem, there is a possibility of converting
the model from nonconvex to convex by some recent researches. In addition, for some
problems that cannot be solved by mathematical formulation, some non-numerical
procedures could be successfully applied.
(3) What the top issue of numerical calculation is that an approximate solution with
engineering precision can be obtained, by which the difficulty of the optimization model
could be tested under the conditions without need of an analytical solution, and the
consistency between the theoretical model and actual problem could be verified.
However, because any numerical calculation method has its own limitation, the complex
relationship between a theoretical model and a practical problem may be expressed by
computer vision technology in the near future.
(4) The mutual transformation of mathematical models is very helpful for solving difficult
problems, because mathematical models can be transformed into each other under such
certain conditions, such as discrete and continuous, accurate and approximate, differential
and difference, solvable and nonsolvable, convex and nonconvex, optimal and
suboptimal, etc.

Solvability discussions about mathematical models are also explained in this book, such as
search scope, initial point, the limit of the variables, and the range of the equations. Some
special modeling techniques for practical engineering problems are also provided in this book.
Some ideas are briefly given in the following, including ideas about how to set variables and
functions, ideas about how to determine model types and algorithms. These modeling
techniques have been implemented in the practical problems of this book and can be effectively
applied to help solve power system optimization problems.