Page 117 - Mathematical Models and Algorithms for Power System Optimization

New Algorithms Related to Power Flow 107
k k+1
Step 5: Test for the convergence of the nonlinear OPF problem. If |Z Z | ε (ε is a
small integer), then terminate the calculation procedure, because a near-optimal solution
ZR of the nonlinear discrete OPF problem has already been obtained. Otherwise return to
Step 3 after setting k¼k+1.

0
Note that the final result is relevant to Z . The OPF is an operation problem, and the operation
bus will not be too far from the optimal solution. Thus, processing the variation of the limit
values during the solution process will satisfactorily solve the nonlinear problem. The method
is shown in Section 4.7.3 in detail.

4.7.2 Linearization of the Problem

To formulate the mixed-integer LP problem of the discrete OPF, the original problem can be
k
linearized by the Taylor series expansion around Z . The Jacobian matrix shown in the next
section can be used as a coefficient for the continuous variable vector and integer variable
vector of the mixed-integer LP problem shown here. To simplify, the superscript k for the
variable in the formula is omitted.
ι
ι
min c X + d Y (4.31)
s:t: AX + BY ¼ b (4.32)

X X X (4.33)

Y Y Y (4.34)

where X—continuous variable; Y—integer variable; A, B—constant coefficient matrix; b, c,
d—constant coefficient vector.

Eq. (4.31) can be expressed as ∂f i (P Gi )/∂X i +∂f i (P Gi )/∂Y i , where coefficients c and d are
combinations of coefficients such as a 1i and a 2i . The method for obtaining the coefficient of the
linear equation for the nondifferentiable integer variable Y can be approximated by the
difference expression, as shown in Appendix B.

4.7.3 SLP-based Solution Procedure for Linear MIP Problem

The SLP technique is used to calculate the next solution after the linear solution of the LP
problem and the limits of variables have been obtained. In this method, the limits of the
variable have to be changed to satisfy the given criteria. If such criteria are satisfied, increase
the limits; otherwise, decrease the limits. As previously explained, in Step 4 of the general
computational procedure for discrete OPF, the approximate mixed-integer LP method in
k
Appendix A can be used to search for the integer feasible solution Z of the mixed-integer
LP problem. The limits of the variables also need to be determined and adjusted, that is,