Page 193 - Mathematical Models and Algorithms for Power System Optimization
P. 193
184 Chapter 6
Based on the situation, one unit of infeasibility degree (j¼1, 2) is added and defined as follows:
q i1 ¼ ð½ f a 1 d 11 Þ x 1 + x 2 a 2 d 12 Þ½ ð g
(6.39)
q i2 ¼ ð½ f a 1 d 21 Þ x 1 +0g
Therefore, as shown in Eq. (6.39), the value of q i1 is the sum of infeasibilities when the value of
y 1 in state-i is increased by one. Assuming the condition in Eq. (6.38) also holds, the feasibility
measure h ij is defined as:
ð
ð
h i1 ¼ min a 1 xð½ 1 Þ= d 11 Þ, x 2 a 2 Þd 12
(6.40)
h i2 ¼ min a 1 xð½ 1 Þ= d 21 Þ, a 2 x Þ= d 22 Þ
ð
ð
ð
2
It may be seen from Eq. (6.40) that the value of h i1 means that the value of y 1 can be
decreased by h i1 without making state-i infeasible. Eqs. (6.39)and 6.40) give the simple case
with only two integers and two basic variables. This is the foundation for this section to
introduce the concepts of multistate infeasibility q ij and feasibility h ij , which will then be
used to consider for the coordination procedure.
To illustrate the practical sense of q ij and h ij , as well as decomposition and coordination
procedure, in Fig. 6.5, q ij and h ij in each subproblem i represent the infeasibility measure and
feasibility measure, respectively. These measures are used to determine the integer variable Y cj
to be adjusted in the linearized master problem (LMP). A simple two-state decomposition and
coordination procedure is used to illustrate the concept of decomposition and coordination in
Fig. 6.6.
Based on this preparation, the decomposition and coordination procedure is shown in Fig. 6.5.
Given in the following text is the detailed procedure step by step.
Substep 0: The number of existing and newly installed VAR equipment is determined by
the following procedures:
(1) If iteration counter k ¼0, the integer constraints are relaxed, and each subproblem
can be solved with LP, then the obtained solution shall be rounded off, and Y ij will be
used to determine the upper limit of coordinate variable, assuming that:
Y Cj ¼ max Y ij , j 2 N (6.41)
i2M
(2) If the iteration count k >0, the integer solution from the previous iteration may be
taken as the initial integer solution for this iteration.
Substep 1: Each subproblem with integer constraints is solved by using the approximation
method in Section 6.2, where the value of Y C is taken as the upper limit of the number of
VAR source unit installation. If all of the subproblems (i ¼1, 2, …, N) are feasible, then go
to Substep 4. Otherwise, go to Substep 2.
Substep 2: For each of the infeasible states, infeasibility measure q ij is calculated, and
infeasibility measure q j , when Y cj is increased by one, is defined as:
