Use of Linear and Quadratic Programming Techniques in Exploiting the Nonlinear Features of the Optimal Power Flow

Abstract

The ordinary power flow and the optimal power flow due to the nature of the problem have several nonlinear features. The important ones are related to the nodal power specifications and to the formulation of the objective function. Since there are many inequality constraints there is a strong tendency to linearize the problem and to apply LP and QP-techniques in order to achieve an efficient solution process. Any linearization, however, will increase the computational effort because of an increase in the number of iterations. Quadratic approximations adapt in a better way, however, they require more effort in the solution process. In application programs special approaches have been taken to exploit the well-known features of LP and QP. A special subject within the optimal power flow is the treatment of the objective function. When separability of the nonlinear conditions is given the process of linearization can be adapted in such as way that segments of varying sizes can be used to account for the prevailing accuracy. The LP-solution process is confined to a small number of segments thereby keeping the number of variables small. As the accuracy increases the size of the segments is reduced. An integrated LP method with a control process for the selection of segments is possible.

The second approach takes advantage of the basic property of a quadratic programming problem that it can be reduced to two linear problems, one solving the unconstrained optimality conditions and the other the inequality constraints. The latter is an LP problem. As it turns out the reduction constitutes the dual quadratic problem to the original formulation. In order to achieve satisfactory performance special measures have to be taken, e.g. to exploit the sparsity of the problem. As far as the nonlinearities of the load flow problem itself is concerned a practical approach is the linearization applied in each iteration. As a consequence a quadratic problem results which is solved by linear means. Formulations of the problem vary with the tratement of the sparsity of the system. Examples of solved problems will be given.