Page 119 - Mathematical Models and Algorithms for Power System Optimization

New Algorithms Related to Power Flow 109
k
where x j is the continuous variable in the iteration k. The following constraint is given to each
integer variable y j of the constraints in the linear MIP problem:

k
E y j y E, E 0,EE is an integer (4.36)
j
k
where y j is the discrete variable of the iteration k.
Substep 2: If the bounds S and E are given, the approximate mixed-integer LP method
in Appendix A is used to obtain a near-optimal solution of the linear MIP problem in
which the bound constraints, Eqs. (4.35) and (4.36), are added. The obtained near-optimal
k+1
solution is denoted by Z .
k+1 k
Substep 3: Check whether the maximum infeasibility amount of Z relative to Z has been
decreased, that is, whether the following condition is satisfied:

k
k +1
MAXINF Z > MAXINF Z (4.37)
where MAXINF(Z) is defined as:

f
MAXINF ZðÞ ¼ max j g i X, Yð Þ u i j g (4.38)
where g i (X,Y) is the power flow constraint function, and u i is the unbalance amount produced
after the solution to the LP problem is substituted into the nonlinear power flow constraint
function. That is, Eq. (4.38) means the maximum calculation error of PQ bus or the maximum
violation value of the active power and reactive power of generator bus after the linearized
solutions X and Y are substituted into the original nonlinear power flow constraint Eqs. (4.11)
k
k
and (4.12). In the case of MAXINF(Z )¼0or MAXINF(Z )¼ MAXINF(Z k+1 ), there is no
k
violation value or the violation values in the two iterations are the same, then MAXINF(Z )
cannot be used as the criteria. Instead, the change of objective function values between the two
iterations is added to the criteria, shown in this formula:
k k
k
MAXINF Z + fX , Y (4.39)
k
k
where the definition of f(X , Y ) is given in Eq. (4.27). If the conditions for Eq. (4.37) are
satisfied, Z k+1 is accepted and the iterative procedure for mixed-integer LP problem terminates;
otherwise, the next step begins to calculate Z k+1 again.
Substep 4: To reduce infeasibility amount, adjust step bounds S and E. If bounds S and E
can be further reduced, reduce them. For example, set S S/2, E E 1 (if E >0) and
return to Substep 1. If the bounds S and E are already small enough, the calculation process
terminates. At this time, vector Z cannot be further improved, but it has been forced to
converge.

The necessity of the bound constraints in the optimization procedure is discussed here. The SLP
technique always converges to a nonlinear solution by controlling the variation range of the