Page 120 - Mathematical Models and Algorithms for Power System Optimization

110 Chapter 4

bound for the variables during the iterative process. Therefore, in the initial step of iteration,
the variation range of the variables must be determined. In Substep 2 and the subsequent
iteration process, the bounds S and E will be changed in Substep 4, and the principle of such
change will depend on the nonlinear solutions within the two iterations. If the nonlinear
infeasibility amount increases, the values of the bound are reduced by half; otherwise, the
values of the bound remain unchanged. The iterative process is to search for the solution
under the linear constraint, although the actual constraint is nonlinear. Therefore, in Substep 3
of the iteration, it is necessary to check whether the violation of the nonlinear constraint
is improved.
During the entire solution procedure, because the linearized objective function of Eq. (4.31) is
optimized under the linearized constraints of Eqs. (4.32)–(4.34), there is no guarantee that the
k
resulting solution Z k+1 is better than Z , that is, Z k+1 may violate the nonlinear constraints of
Eq. (4.28) more seriously and with a higher value of objective function. Therefore, after
obtaining Z k+1 in the optimization procedure of the linear MIP problem, it must be checked
k
whether Z k+1 is better than solution Z from the previous iteration.
In this algorithm, Z k+1 is considered acceptable if it makes the infeasibility of the nonlinear
equations smaller. If the foregoing criterion is not satisfied, then the linear MIP problem must
be resolved after reducing the bounds S and E (the bounds are within the constraints of the
variables X and Y). The reason is that the linearized accuracy is related to the variation of the
k k+1
variable vector Z between two iterations. If the variable range becomes very small, jZ Z j
will become smaller, thus, the variable Z converges.

4.8 Implementation of Discrete OPF

4.8.1 Verification by the Concrete Formulation of 5-Bus System
(1) Original data: The 5-bus system is as shown in Fig. 4.6. This system has two
adjustable transformer tap ratios and three capacitors. The detailed data are given in
Tables 4.9–4.12.

Fig. 4.6

5-Bus system diagram.